# Total and Local Quadratic Indices of the Molecular Pseudograph's Atom Adjacency Matrix: Applications to the Prediction of Physical Properties of Organic Compounds

## Abstract

**:**

^{i}spaces. In this way we can represent molecules having a total of i atoms as elements (vectors) of the vector spaces ℜ

^{i}(i=1, 2, 3,..., n; where n is number of atoms in the molecule). In these spaces the components of the vectors are atomic properties that characterize each kind of atom in particular. The total quadratic indices are based on the calculation of mathematical quadratic forms. These forms are functions of the k-th power of the molecular pseudograph’s atom adjacency matrix (M). For simplicity, canonical bases are selected as the quadratic forms’ bases. These indices were generalized to “higher analogues” as number sequences. In addition, this paper also introduces a local approach (local invariant) for molecular quadratic indices. This approach is based mainly on the use of a local matrix [M

^{k}(G, F

_{R})]. This local matrix is obtained from the k-th power (M

^{k}(G)) of the atom adjacency matrix M. M

^{k}(G, F

_{R}) includes the elements of the fragment of interest and those that are connected with it, through paths of length k. Finally, total (and local) quadratic indices have been used in QSPR studies of four series of organic compounds. The quantitative models found are significant from a statistical point of view and permit a clear interpretation of the studied properties in terms of the structural features of molecules. External prediction series and cross-validation procedures (leave-one-out and leave-group-out) assessed model predictability. The reported method has shown similar results, compared with other topological approaches. The results obtained were the following: a) Seven physical properties of 74 normal and branched alkanes (boiling points, molar volumes, molar refractions, heats of vaporization, critical temperatures, critical pressures and surface tensions) were well modeled (R>0.98, q

^{2}>0.95) by the total quadratic indices. The overall MAE of 5-fold cross-validation were of 2.11

^{o}C, 0.53 cm

^{3}, 0.032 cm

^{3}, 0.32 KJ/mol, 5.34

^{o}C, 0.64 atm, 0.23 dyn/cm for each property, respectively; b) boiling points of 58 alkyl alcohols also were well described by the present approach; in this sense, two QSPR models were obtained; the first one was developed using the complete set of 58 alcohols [R=0.9938, q

^{2}=0.986, s=4.006

^{o}C, overall MAE of 5-fold cross-validation=3.824

^{o}C] and the second one was developed using 29 compounds as a training set [R=0.9979, q

^{2}=0.992, s=2.97

^{o}C, overall MAE of 5-fold cross-validation=2.580

^{o}C] and 29 compounds as a test set [R=0.9938, s=3.17

^{o}C]; c) good relationships were obtained for the boiling points property (using 80 and 26 cycloalkanes in the training and test sets, respectively) using 2 and 5 total quadratic indices: [Training set: R=0.9823 (q

^{2}=0.961 and overall MAE of 5-fold cross-validation=6.429

^{o}C) and R=0.9927 (q

^{2}=0.977 and overall MAE of 5-fold cross-validation=4.801

^{o}C); Test set: R=0.9726 and R=0.9927] and d) the linear model developed to describe the boiling points of 70 organic compounds containing aromatic rings has shown good statistical features, with a squared correlation coefficient (R

^{2}) of 0.981 (s=7.61

^{o}C). Internal validation procedures (q

^{2}=0.9763 and overall MAE of 5-fold cross-validation=7.34

^{o}C) allowed the predictability and robustness of the model found to be assessed. The predictive performance of the obtained QSPR model also was tested on an extra set of 20 aromatic organic compounds (R=0.9930 and s=7.8280

^{o}C). The results obtained are valid to establish that these new indices fulfill some of the ideal requirements proposed by Randić for a new molecular descriptor.

## Introduction

- Research on drugs, toxics or generally any organic molecules with a common skeleton, which is responsible for the activity or property under study.
- Study of the reactivity of specific sites of a series of molecules, which can undergo a chemical reaction or enzymatic metabolism.
- In the study of molecular properties such as spectroscopic measurements, which are calculated experimentally in a local fashion
- In any general case where it is necessary to study not the molecule as a whole, but rather some local properties of certain fragments, then the definition of local descriptors could be necessary.

**TOMO-COMD (**acronym of

**TO**pological

**MO**lecular

**COM**puter

**D**esign) [36]. It calculates several families of topological molecular descriptors. One of these families has been defined as quadratic indices by analogy with the quadratic mathematical forms.

## Results and Discussion

#### Computational methods. Mathematical definition of the molecular descriptor

Molecular vector space

_{A}) [37] of the atom A take the values X

_{H}= 2.2 for Hydrogen, X

_{C}= 2.63 for Carbon, X

_{N}= 2.33 for Nitrogen, X

_{O}= 3.17 for Oxygen, X

_{Cl}= 3.0 for Chlorine and so on.

_{A}. Thus, a molecule having 2, 3, 4,…, n atoms can be “represented” by means of vectors, with 2, 3, 4,...., n components, belonging to the spaces ℜ

^{2}, ℜ

^{3}, ℜ

^{4},..., ℜ

^{n}, respectively. Where n is the dimension of these real subsets (ℜ

^{n}).

_{C}, X

_{C}, X

_{C}, X

_{C}, X

_{C}, X

_{C}). On the other hand, n-propanol, iso-propanol, propanal, and acetone may be represented by (X

_{C}, X

_{C}, X

_{C}, X

_{O}) or any permutation of the components of this vector. All these vectors belong to the product space ℜ

^{6}and ℜ

^{4}, respectively. It must be noted that the order of the vector components is meaningless here. This fact, not common in classical vector spaces, will be explained elsewhere. In this example the hydrogen atoms were not considered.

**E**) could be defined:

^{k}⌒ ℜ

^{l}= {0}: k ≠ l [38,39] and the dimension of

**E**is the sum of the dimensions of each one of the ℜ

^{i}spaces. Therefore, this dimension is n(n+1)/2.

**spaces. This mathematical formalism makes it possible to represent any drug or organic molecule as a vector space and then, to use the well-known applications of this algebraic construction to codify molecular structure in a timely but mathematically rigorous way.**

^{n}#### Total quadratic indices; [q_{k}(x)].

**q**: H→ K is a quadratic form (

**q**(x)) if for X=x

_{1}a

_{1}+...+x

_{n}a

_{n}, where (a

_{i})

_{1≤i≤}n is a base of H, it satisfies that:

^{i}, vector space of finite dimension i:

**q**: ℜ

^{i}→ K. If a molecule is considered with n atoms (vector of ℜ

^{n}), the k-th quadratic indices

**q**(x) are defined as

_{k}**q**application (

**q**: ℜ

^{n}→ℜ

**)**if the molecular vector (X) can be expressed by a linear combination with a base belonging to the vector space ℜ

^{n}(X=x

_{1}a

_{1}+...+x

_{n}a

_{n}, where (a

_{i})

_{1≤i≤}n is a base of ℜ

^{n}). Taking into consideration the above mentioned conditions

**q**is a quadratic form if Eq. 3 is considered. In this way, the whole form

**q**

_{k}(x), is written as a sum of all the possible terms a

_{ij}x

_{i}x

_{j}, of "i" and "j", independently one of the other, taking values from 1 to n.

^{k}a

_{ij}=

^{k}a

_{ji}and n is the number of atoms of the molecule. The coefficients

^{k}a

_{ij}are the elements of the k-th power of the “molecular pseudograph’s atom adjacency matrix” (G). Here,

**M**(G)

**= M =**[a

_{ij}], where n is the number of vertices and the elements a

_{ij}are defined as follows:

_{ij}is the number of edges that comply with e

_{k}~ v

_{i},v

_{j}among the vertices (atoms) v

_{i}and v

_{j}and L

_{ii}is the number of loops in v

_{i}. Thus, mathematically a pseudograph can be defined in the following way [38,39]: Let V be a finite not empty set and E an unordered finite set of pairs of elements in V (with equal pairs in E inclusive): the pairs G=<V,E >, are called graphs with loops and multiple edges or pseudograph.

_{ij}(if a

_{ij}= P

_{ij}) of this matrix represent the bonds between an atom v

_{i}and an other v

_{j}. The matrix

**M**provides the number of walks of length k that links the vertices v

^{k}_{i}and v

_{j}. For this reason each edge represents 2 electrons of a covalent bond between atoms v

_{i}and v

_{j}, and it is appreciated in the

**M**(k=1) matrix input that v

_{ij}and v

_{ji}is equal to 1. In this way, the benzene molecule can be represented by two different multigraphs, where each multigraph is related with one of the Kekulé structures. Taking this into consideration, it is necessary the use of a pseudograph to avoid this situation in compounds with more than one canonical structure. This happens for substituted aromatic compounds such as pyridine, naphthalene, quinoline, etc., where the electrons of PI(π)-orbitals are represented as loops of all-ring atoms.

**q**(x) the following considerations arise in a natural way: 1) With the coefficients a

_{k}_{ij,}evidently, the square matrix

**M**=[a

_{ij}] of order n can be formed, and 2) let X = [x

_{1}, x

_{2}, x

_{3},...., x

_{n}], the vector of coordinates of X in the base {a

_{1},...,a

_{i}}, a matrix of n-row and a single columns; transposing this matrix, X

^{t}= [X

_{1}X

_{2},........,X

_{n}] is obtained; which is the row vector of the coordinates of X in the base {a

_{1},...,a

_{i}}. Then

**q**(x) can be written in the form of a matrix product

**q**(x) =X

^{t}

**M**X. Recently, other descriptors have been expressed through the vector-matrix-vector multiplication procedure [42]. The result of the matrix multiplication is a matrix formed by a row and a column that is a number. Therefore, if we use the canonical bases, the coordinates of any molecular vector (X) coincide with the components of that vector. For that reason, those coordinates can be considered as weights (atom labels) of the vertices of the molecular pseudograph, due to the fact that components of the vector are values of some atomic property, which characterizes each kind of atom.

**Table 1.**Total and Local Quadratic Indices Calculated for Multigraphs (MKA, MKB) and Pseudographs (P).

Benzene | ||||||||
---|---|---|---|---|---|---|---|---|

q_{0}(x) | q_{1}(x) | q_{2}(x) | q_{3}(x) | q_{4}(x) | q_{5}(x) | q_{6}(x) | q_{7}(x) | |

P | 41.5014 | 124.5042 | 373.5126 | 1120.5378 | 3361.6134 | 10084.8402 | 30254.5206 | 90763.5618 |

MKA | 41.5014 | 124.5042 | 373.5126 | 1120.5378 | 3361.6134 | 10084.8402 | 30254.5206 | 90763.5618 |

MKB | 41.5014 | 124.5042 | 373.5126 | 1120.5378 | 3361.6134 | 10084.8402 | 30254.5206 | 90763.5618 |

Acetylsalicylic acid | ||||||||

q_{0}(x) | q_{1}(x) | q_{2}(x) | q_{3}(x) | q_{4}(x) | q_{5}(x) | q_{6}(x) | q_{7}(x) | |

P | 102.4477 | 268.8912 | 873.5982 | 2566.8034 | 8381.4114 | 25593.6122 | 83330.7872 | 260026.931 |

MKA | 102.4477 | 268.8912 | 873.5982 | 2549.8376 | 8284.7898 | 25063.374 | 81351.7828 | 250745.988 |

MKB | 102.4477 | 268.8912 | 873.5982 | 2566.5118 | 8389.425 | 25513.2092 | 83389.772 | 258104.308 |

^{E}q_{0}(x) | ^{E}q_{1}(x) | ^{E}q_{2}(x) | ^{E}q_{3}(x) | ^{E}q_{4}(x) | ^{E}q_{5}(x) | ^{E}q_{6}(x) | ^{E}q_{7}(x) | |

P | 40.1956 | 58.3597 | 265.963 | 510.2749 | 2171.4817 | 4947.1654 | 19328.9482 | 49869.8377 |

MKA | 40.1956 | 58.3597 | 265.963 | 500.226 | 2133.2198 | 4618.7534 | 18773.2472 | 44486.7656 |

MKB | 40.1956 | 58.3597 | 265.963 | 508.5631 | 2201.8503 | 4802.1696 | 19870.6695 | 47162.9747 |

^{H}q_{0}(x) | ^{H}q_{1}(x) | ^{H}q_{2}(x) | ^{H}q_{3}(x) | ^{H}q_{4}(x) | ^{H}q_{5}(x) | ^{H}q_{6}(x) | ^{H}q_{7}(x) | |

P | 4.84 | 6.974 | 10.626 | 33.682 | 67.54 | 270.578 | 670.604 | 2600.972 |

MKA | 4.84 | 6.974 | 10.626 | 33.682 | 67.54 | 269.632 | 647.306 | 2589.686 |

MKB | 4.84 | 6.974 | 10.626 | 33.682 | 67.54 | 271.766 | 653.092 | 2639.868 |

Metolazone | ||||||||

q_{0}(x) | q_{1}(x) | q_{2}(x) | q_{3}(x) | q_{4}(x) | q_{5}(x) | q_{6}(x) | q_{7}(x) | |

P | 171.9119 | 485.942 | 1711.0469 | 5439.1693 | 19235.232 | 62338.8312 | 220106.56 | 721470.089 |

MKAA | 171.9119 | 485.942 | 1711.0469 | 5424.1812 | 19161.672 | 61839.7906 | 218582.941 | 710431.996 |

MKAB | 171.9119 | 485.942 | 1711.0469 | 5411.9254 | 19107.9148 | 61560.958 | 217543.348 | 706114.062 |

MKBA | 171.9119 | 485.942 | 1711.0469 | 5426.3854 | 19199.863 | 61837.827 | 219141.462 | 710613.352 |

MKBB | 171.9119 | 485.942 | 1711.0469 | 5414.1296 | 19146.1058 | 61558.9944 | 218101.869 | 706307.674 |

^{E}q_{0}(x) | ^{E}q_{1}(x) | ^{E}q_{2}(x) | ^{E}q_{3}(x) | ^{E}q_{4}(x) | ^{E}q_{5}(x) | ^{E}q_{6}(x) | ^{E}q_{7}(x) | |

P | 61.2415 | 133.8902 | 554.1099 | 1558.9199 | 6272.0672 | 18784.7951 | 73539.8425 | 228597.096 |

MKAA | 61.2415 | 133.8902 | 554.1099 | 1545.5098 | 6202.9256 | 18310.0294 | 72577.097 | 218343.795 |

MKAB | 61.2415 | 133.8902 | 554.1099 | 1539.3819 | 6196.7977 | 18225.9483 | 72439.9618 | 217339.95 |

MKBA | 61.2415 | 133.8902 | 554.1099 | 1553.8419 | 6260.6838 | 18444.8521 | 73549.9487 | 220551.513 |

MKBB | 61.2415 | 133.8902 | 554.1099 | 1547.714 | 6254.5559 | 18360.771 | 73412.8135 | 219553.796 |

^{H}q_{0}(x) | ^{H}q_{1}(x) | ^{H}q_{2}(x) | ^{H}q_{3}(x) | ^{H}q_{4}(x) | ^{H}q_{5}(x) | ^{H}q_{6}(x) | ^{H}q_{7}(x) | |

P | 14.52 | 15.378 | 46.376 | 146.608 | 380.556 | 1654.686 | 4353.734 | 19526.76 |

MKAA | 14.52 | 15.378 | 46.376 | 146.608 | 381.216 | 1662.65 | 4285.534 | 19850.446 |

MKAB | 14.52 | 15.378 | 46.376 | 146.608 | 381.216 | 1662.65 | 4284.588 | 19835.926 |

MKBA | 14.52 | 15.378 | 46.376 | 146.608 | 380.27 | 1647.096 | 4238.41 | 19605.3 |

MKBB | 14.52 | 15.378 | 46.376 | 146.608 | 380.27 | 1647.096 | 4237.464 | 19590.78 |

Prazocin | ||||||||

q_{0}(x) | q_{1}(x) | q_{2}(x) | q_{3}(x) | q_{4}(x) | q_{5}(x) | q_{6}(x) | q_{7}(x) | |

P | 198.7612 | 541.9074 | 1696.6156 | 5358.4782 | 17314.5582 | 56186.8214 | 183864.863 | 603661.363 |

MKAA | 198.7612 | 541.7274 | 1694.1796 | 5323.0646 | 17197.7804 | 55637.9444 | 181811.302 | 595116.828 |

MKAB | 198.7612 | 541.7274 | 1694.3596 | 5327.7986 | 17244.174 | 55914.3384 | 183221.047 | 601548.719 |

MKBB | 198.7612 | 541.7274 | 1694.3596 | 5335.6406 | 17224.5402 | 55735.215 | 181942.392 | 595274.105 |

^{E}q_{0}(x) | ^{E}q_{1}(x) | ^{E}q_{2}(x) | ^{E}q_{3}(x) | ^{E}q_{4}(x) | ^{E}q_{5}(x) | ^{E}q_{6}(x) | ^{E}q_{7}(x) | |

P | 67.3401 | 144.9615 | 468.8527 | 1384.3378 | 4526.6829 | 14281.5586 | 46761.2533 | 151360.249 |

MKAA | 67.3401 | 146.3595 | 475.5165 | 1381.8781 | 4632.9291 | 14424.8713 | 48134.0569 | 153961.075 |

MKAB | 67.3401 | 146.3595 | 474.1185 | 1363.4944 | 4559.3158 | 14146.1775 | 47209.3348 | 151083.318 |

MKBB | 67.3401 | 146.3595 | 474.1185 | 1377.4643 | 4553.9629 | 14140.7919 | 46743.0601 | 149152.807 |

^{H}q_{0}(x) | ^{H}q_{1}(x) | ^{H}q_{2}(x) | ^{H}q_{3}(x) | ^{H}q_{4}(x) | ^{H}q_{5}(x) | ^{H}q_{6}(x) | ^{H}q_{7}(x) | |

P | 9.68 | 10.252 | 30.932 | 64.152 | 216.128 | 645.392 | 2236.476 | 7512.296 |

MKAA | 9.68 | 10.252 | 30.932 | 64.152 | 220.088 | 668.8 | 2359.72 | 7965.76 |

MKAB | 9.68 | 10.252 | 30.932 | 62.832 | 208.516 | 616.484 | 2135.1 | 7120.168 |

MKBB | 9.68 | 10.252 | 30.932 | 62.832 | 208.516 | 615.912 | 2111.956 | 7031.288 |

**M**the matrix of paths of length k (

**M**) among n vertices of the molecular pseudograph and we multiply it by the coordinates of molecular vector (X) in the canonical basis of ℜ

^{k}^{n}, we obtain k values that constitute numeric descriptors of the molecular structure. Therefore we can “define” a molecule as quadratic indices (

**q**(x)’s) in the matrix form X

^{t}

**M**

^{k}X =

**q**

_{k}(x), k ≥ 10.

**M**and

**q**

_{k}(x) it can be observed that the total quadratic indices are positive integers. The data presented in Table 2 exemplifies the calculation of five quadratic indices for isonicotinic acid.

**Table 2.**Definition and Calculation of Five (k=0-4) Quadratic Indices of the Molecular Pseudograph’s Atom Adjacency Matrix of the Isonicotinic Acid Molecule.

Isonicotinic acid MolecularStructure | Molecular Pseudograph (G)(Hydrogen Suppressed-pseudograph) | X=[N1 C2 C3 C4 C5 C6 C7 O8 O9]Molecular Vector: X∊ℜ^{9} and ℜ^{9}∊E;E: Molecular Vector SpaceIn the definition of the X, as molecular vector, the chemical symbol of the element is used to indicate the corresponding electronegativity value. That is: if we write O it means χ(O), oxygen Mulliken electronegativity or some atomic property, which characterizes each atom in the molecule. Therefore, if we use the canonical bases of R^{9}, the coordinates of any vector X coincide with the components of that molecular vectorX ^{t} =[233 263 263 263 263 263 263 3.17 3.17]X ^{t} = transposed of X and it means the vector of the coordinates of X in the Canonical basis of R^{9} (a row vector)X: vector of coordinates of X in the Canonical basis of R^{9} (a column vector) | |

$${q}_{0}(x)=\sum _{i=1}^{n}\text{}\sum _{j=1}^{n}{}_{0}\text{}{a}_{ij}{X}_{i}{X}_{j}$$
| = X^{t}M^{0}X=67.0281 | ||

$${q}_{1}(x)=\sum _{i=1}^{n}\text{}\sum _{j=1}^{n}{}_{1}\text{}{a}_{ij}{X}_{\text{}i}{X}_{\text{}j}$$
| = X^{t}M^{1}X=183.7166 | ||

$${q}_{2}(x)=\sum _{i=1}^{n}\text{}\sum _{j=1}^{n}{}_{2}\text{}{a}_{ij}{X}_{\text{}i}{X}_{\text{}j}$$
| = X^{t}M^{2}X=589.963 |
$$\mathrm{M}(G)=\begin{array}{cccccccccc}\text{}& N1& C2& C3& C5& 1& C6& C7& C8& C9\\ N1& 1& 1& 0& 0& 0& 1& 0& 0& 0\\ C2& 1& 1& 1& 0& 0& 0& 0& 0& 0\\ C3& 0& 1& 1& 1& 0& 0& 0& 0& 0\\ C4& 0& 0& 1& 1& 1& 0& 1& 0& 0\\ C5& 0& 0& 0& 1& 1& 1& 0& 0& 0\\ C6& 1& 0& 0& 0& 1& 1& 0& 0& 0\\ C7& 0& 0& 0& 1& 0& 0& 0& 2& 1\\ C8& 0& 0& 0& 0& 0& 0& 2& 0& 0\\ C9& 0& 0& 0& 0& 0& 0& 1& 0& 0\end{array}$$
M(G): Adjacency Matrix Among Vertices of the Molecular Pseudograph (G) | |

$${q}_{3}(x)=\sum _{i=1}^{n}\text{}\sum _{j=1}^{n}{}_{3}\text{}{a}_{ij}{X}_{\text{}i}{X}_{\text{}j}$$
| = X^{t}M^{3}X=1784.6905 | ||

$${q}_{4}(x)=\sum _{i=1}^{n}\text{}\sum _{j=1}^{n}{}_{4}\text{}{a}_{ij}{X}_{\text{}i}{X}_{\text{}j}$$
| = X^{t}M^{4}X=5707.7232 |

#### Local quadratic indices; [q_{kL}(x)]

_{R}(connected subgraph), within a specific pseudograph (G) is the following:

^{k}

**a**

_{ijL}is the element of the file “i” and column “j” of the matrix

**M**=

^{k}_{L}**M**(G, F

^{k}_{R}) [

**q**

_{kL}(x) =

**q**

_{k}(x, F

_{R})]. This matrix is extracted from the

**M**matrix and it contains the information referred to the vertices of the specific fragments (F

^{k}_{R}) and also of the molecular environment.

**M**

^{k}_{L}=[

^{k}a

_{ijL}] with elements

^{k}a

_{ijL}is defined as follows:

^{k}a

_{ijL}=

^{k}a

_{ij}if both v

_{i}and v

_{j}are vertices contained in the specific fragment.

=1/2

^{k}a

_{ij}either v

_{i}or v

_{j}is contained in the specific fragment but not both

at the same time

=0 otherwise

^{k}a

_{ij}being the elements of the k-th power of

**M**. These local analogues can also be expressed in matrix form by the expression:

_{kL}(x) =X

^{t}

**M**X:

^{k}_{L}**M**

^{k}_{L}:it is extract from

**M**

^{k}

**M**can be partitioned in Z local matrices

^{k}**M**L=1,... Z. The k-th power of matrix

^{k}_{L},**M**is exactly the sum of the k-th power of local Z matrices:

**M**=[

^{k}^{k}a

_{ij}], where:

_{R}

_{R}is contained. High values of k are in relation to the environment information of the fragment F

_{R}considered inside the molecular pseudograph (G). A general equation for k order is described as follows:

#### Calculation of total and local quadratic indices

- 1)
- Total and Local indices of zero order [
**q**(x) and_{0}**q**(x)]. These indices are obtained when the matrix_{0L}**M**is raised to the power 0 (k=0). A matrix raised to the power 0 is the identity matrix (**I**); which is constituted by the elements a_{ii}=1 [**M**^{0}(i, i)=1]. Since the zero order matrix is diagonal, its quadratic form contains only the terms with the squares of the coordinates (an atomic property) of the X vector in canonical bases. Generally, we can establish that.$${q}_{0}(x)=\sum _{i=1}^{n}X{i}^{2}$$$${q}_{0L}(x)=\sum _{i=1}^{m}X{i}^{2}$$_{R}under study, respectively.

**q**

_{0}(x)=X

^{t}

**M**X and local quadratic indices of zero order for each one of the three represented fragments are calculated using the three local matrices as the matrix of the quadratic form. Making the matrix product by the row matrix (X

^{0}^{t}) and by the column matrix (X), the three local molecular quadratic indices (one for each fragment) are obtained (see Table 3):

**q**(x,

_{0}**F**

_{1})=1

^{.}(X

_{O4 })

^{2}=1

^{.}(3.17)

^{2}=10.0489;

**q**(x,

_{0}**F**

_{2})= 1

^{.}(X

_{C3})

^{2}+ 1

^{.}(X

_{C5})

^{2}=1

^{.}(2.63)

^{2 }+1

^{.}(2.63)

^{2}=13.8338 and

**q**(x,

_{0}**F**

_{3})= 1

^{.}(X

_{C1})

^{2}+ 1

^{.}(X

_{C2})

^{2}=1

^{.}(2.63)

^{2 }+1

^{.}(2.63)

^{2}=13.8338. It should be noted that

**q**(x,

_{0}**G**)=

**q**(x,

_{0}**F**

_{1})+

**q**(x,

_{0}**F**

_{2})+

**q**(x,

_{0}**F**

_{3})= 1

^{.}(X

_{C1})

^{2}+1

^{.}(X

_{C2})

^{2}+1

^{.}(X

_{C3})

^{2}+1

^{.}(X

_{O4})

^{2}+1

^{.}(X

_{C5})

^{2}=1

^{.}(2.63)

^{2 }+1

^{.}(2.63)

^{2 }+1

^{.}(2.63)

^{2}+ 1

^{.}(3.17)

^{2 }+ 1

^{.}(2.63)

^{2}=37.7165 and that

**M**(G)=

^{0}**M**G,

^{0}(**F**

_{1}

**)+M**G,

^{0}(**F**

_{2}

**)+M**G,

^{0}(**F**

_{3}

**).**

**q**(x) contains information about the fragment under study, without regard to which atom(s) it is bonded to, since the ones in the main diagonal express that paths of length 0 is the succession of a single vertex. That is to say, those sub-graphs of zero order consist of isolated vertices. This index has information about the molecular size of the fragment and it depends on the number and type of atoms that are contained in the fragment under study.

_{0L}- 2)
- Total and local quadratic indices of first order [
**q**(x) and_{1}**q**(x)]_{1L}**.**These indices are obtained when the matrix**M**is raised to the unit power (**M**=^{1}**M**) and multiplied by the matrices X^{t}and X. We can write the expression for**q**(x) and_{1}**q**(x) in the forms:_{1L}$${q}_{1}(x)=\sum _{i}{a}_{ii}{{X}_{i}}^{2}+2\sum _{(i,j)}{a}_{ij}{X}_{i}{X}_{j}$$$${q}_{1L}(x)=\sum _{i}{a}_{iiL}{{X}_{i}}^{2}+2\sum _{(i,j)}{a}_{ijL}{X}_{i}{X}_{j}$$

**q**(x)= 4

_{1}^{.}(X

_{C}

_{1}

^{.}X

_{C}

_{2}) + 2

^{.}(X

_{C}

_{2}

^{.}X

_{C}

_{3}) + 2

^{.}(X

_{C}

_{3}

^{.}X

_{O}

_{4}) + 2

^{.}(X

_{C}

_{3}

^{.}X

_{C}

_{5}) = 4

^{.}(2.63

^{.}2.63) +2

^{.}(2.63

^{.}2.63) +2

^{.}(2.63

^{.}3.17) +2

^{.}(2.63

^{.}2.63) = 72.0094. To obtain the local analogues for each fragment we proceed to the extract of the matrices “partitioned” for each one of the fragments (see Table 3). Making the matrix product we get:

**q**(x,

_{1}**F**

_{1}) = 1

^{.}(X

_{C}

_{3}

^{.}X

_{O}

_{4}) = 1

^{.}(2.63

^{.}3.17) = 8.3371;

**q**(x,

_{1}**F**

_{2}) = 1

^{.}(X

_{C}

_{2}

^{.}X

_{C}

_{3}) +1

^{.}(X

_{C}

_{3}

^{.}X

_{O}

_{4})+2

^{.}(X

_{C}

_{3}

^{.}X

_{C}

_{5}) = 1

^{.}(2.63

^{.}2.63) +1

^{.}(2.63

^{.}3.17) +2

^{.}(2.63

^{.}2.63) = 29.0878 and

**q**(x,

_{1}**F**

_{3}) = 4

^{.}(X

_{C}

_{1}

^{.}X

_{C2}) +1

^{.}(X

_{C}

_{2}

^{.}X

_{C}

_{3}) = 4

^{.}(2.63

^{.}2.63) +1

^{.}(2.63

^{.}2.63) = 34.5845. It should be observed that

**q**(x,

_{1}**G**)=

**q**(x,

_{1}**F**

_{1})+

**q**(x,

_{1}**F**

_{2}) +

**q**(x,

_{1}**F**

_{3}) and that

**M**(G)= $\sum _{R=1}^{3}$

^{1}**M**G,

^{1}(**F**

_{R})._{R}of interest, but also has information about the atoms to which this fragment is connected to by a step (by means of a walk of length 1). As it is appreciated from its formulation that this index is capable of differentiating between saturated and unsaturated sub-structures (fragments) inside a molecular pseudograph (molecule). Two sub-graphs will only have the same value, if and only if, both fragments present the same composition, equal topological arrangements among the atoms that constitute them and, the fragments are connected to the same atoms that are not part of the fragment by a path of length 1 (in a step).

- 3)
- Total and local quadratic indices of second order [
**q**(x) and_{2}**q**(x)]. In general, these indices are calculated as:_{2L}$${q}_{2}(x)=\sum _{i=1}^{n}\sum _{j=1}^{n}{\text{}}^{2}{a}_{ij}{X}_{i}{X}_{j}$$$${q}_{2L}(x)=\sum _{i=1}^{m}\sum _{j=1}^{m}{\text{}}^{2}{a}_{ijL}{X}_{i}{X}_{j}$$

**M**

^{2}, which are given in Table 3. If in the four cases (total and three local ones) we carry out the matrix product we obtain:

**q**(x,

_{2}**G**)=4

^{.}(X

_{C}

_{1})

^{2}+5

^{.}(X

_{C}

_{2})

^{2}+3

^{.}(X

_{C}

_{3})

^{2}+1

^{.}(X

_{O}

_{4})

^{2}+1

^{.}(X

_{C}

_{5})

^{2}+4

^{.}(X

_{C}

_{1}

^{.}X

_{C}

_{3})+2

^{.}(X

_{C}

_{2}

^{.}X

_{O}

_{4})+2

^{.}(X

_{C}

_{2}

^{.}X

_{C}

_{5})+2

^{.}(X

_{O}

_{4}

^{.}X

_{C}

_{5})=4

^{.}(2.63)

^{2}+5

^{.}(2.63)

^{2}+3

^{.}(2.63)

^{2}+1

^{.}(3.17)

^{2}+1

^{.}(2.63)

^{2}+4

^{.}(2.63

^{.}2.63) +2

^{.}(2.63

^{.}3.17) +2

^{.}(2.63

^{.}2.63) +2

^{.}(3.17

^{.}2.63)=174.8184;

**q**(x,

_{2}**F**

_{1})=1

^{.}(X

_{C}

_{2}

^{.}X

_{O}

_{4})+1

^{.}(X

_{O}

_{4}

^{.}X

_{C}

_{5})+1

^{.}(X

_{O}

_{4})

^{2}=1

^{.}(2.63

^{.}3.17) +1

^{.}(3.17

^{.}2.63)

+1

^{.}(3.17)

^{2}=26.7231;

**q**(x,

_{2}**F**

_{2})=2

^{.}(X

_{C}

_{1}

^{.}X

_{C}

_{3}) +1

^{.}(X

_{C}

_{2}

^{.}X

_{C}

_{5}) +1

^{.}(X

_{C}

_{4}

^{.}X

_{C}

_{5}) +3

^{.}(X

_{C}

_{3})

^{2}+1

^{.}(X

_{C}

_{5})

^{2}=2

^{.}(2.63

^{.}2.63)

+1

^{.}(2.63

^{.}2.63) +1

^{.}(3.17

^{.}2.63) +3

^{.}(2.63)

^{2}+ 1

^{.}(2.63)

^{2}=56.7554, and

**q**(x,

_{2}**F**

_{3})=2

^{.}(X

_{C}

_{1}

^{.}X

_{C}

_{3}) +1

^{.}(X

_{C}

_{2}

^{.}X

_{C}

_{4}) +1

^{.}(X

_{C}

_{2}

^{.}X

_{C}

_{5}) +4

^{.}(X

_{C}

_{1})

^{2}+5

^{.}(X

_{C}

_{2})

^{2}=2

^{.}(2.63

^{.}2.63)

+1

^{.}(2.63

^{.}3.17) +1

^{.}(2.63

^{.}2.63) +4

^{.}(2.63)

^{2}+5

^{.}(2.63)

^{2}=91.3399.

**q**(

_{2}**x**,

**G**) =

**q**(

_{2}**x**,

**F**

_{1})+

**q**(x,

_{2}**F**

_{2})+

**q**(x,

_{2}**F**

_{3}) and that

**M**(G)=$\sum _{R=1}^{3}$

^{2}**M**G,

^{2}(**F**

_{R}).**Table 3.**The Zero, First and Second Powers of the Molecular “pseudograph’s” Atom Adjacency Matrix and Local Matrices for These Order of Each One of 3 Fragments Shown in the Molecule of 1-methylallyl alcohol (but-3-en-2-ol).

Molecular Structure of 1-methylallyl alchohol (But-3-en-2-ol) | X=[C_{1} C_{2} C_{3} O_{4} C_{5}] Molecular Vector: X∊ℜ^{5} and ^{5}∊ℜE;E: Molecular Vector SpaceIn the definition of the X, as molecular vector, the chemical symbol of the element is used to indicate the corresponding electronegativity value. That is: if we write O it means χ(O), oxygen Mulliken electronegativity or some atomic property, which characterizes each atom in the molecule. Therefore, if we use the canonical bases of ℜ^{5}, the coordinates of any molecular vector X coincide with the components of that molecular vector.X ^{t} = [2.63 2.63 2.63 3.17 2.63]X ^{t} = transposed of X and it means the vector of the coordinates of X in the Canonical basis of ℜ^{5} (a row vector)X: vector of coordinates of X in the Canonical basis of ℜ^{5} (a column vector) | ||||

The zero, first and second powers of the molecular “pseudograph’s” total atom adjacency matrix. | |||||

${M}^{0}(G)=I(G)=\begin{array}{ccccc}1& 0& 0& 0& 0\\ 0& 1& 0& 0& 0\\ 0& 0& 1& 0& 0\\ 0& 0& 0& 1& 0\\ 0& 0& 0& 0& 1\end{array}$ | ${M}^{1}(G)=\begin{array}{ccccc}0& 2& 0& 0& 0\\ 2& 0& 1& 0& 0\\ 0& 1& 0& 1& 1\\ 0& 0& 1& 0& 0\\ 0& 0& 1& 0& 0\end{array}$ | ${M}^{2}(G)=\begin{array}{ccccc}4& 0& 2& 0& 0\\ 0& 5& 0& 1& 1\\ 2& 0& 3& 0& 0\\ 0& 1& 0& 1& 1\\ 0& 1& 0& 1& 1\end{array}$ | |||

The zero, first and second powers of the molecular “pseudograph’s” local atom adjacency matrix of each one of 3 fragments shown in the molecule of 1-methylallyl alcohol | |||||

${M}^{0}(G,{F}_{1})=\begin{array}{ccccc}0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0\\ 0& 0& 0& 1& 0\\ 0& 0& 0& 0& 0\end{array}$ | ${M}^{1}(G,{F}_{1})=\begin{array}{ccccc}0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0\\ 0& 0& 0& 1/2& 0\\ 0& 0& 1/2& 0& 0\\ 0& 0& 0& 0& 0\end{array}$ | ${M}^{2}(G,{F}_{1})=\begin{array}{ccccc}0& 0& 0& 0& 0\\ 0& 0& 0& 1/2& 0\\ 0& 0& 0& 0& 0\\ 0& 1/2& 0& 1& 1/2\\ 0& 0& 0& 1/2& 0\end{array}$ | |||

${M}^{0}(G,{F}_{2})=\begin{array}{ccccc}0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0\\ 0& 0& 1& 0& 0\\ 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 1\end{array}$ | ${M}^{1}(G,{F}_{2})=\begin{array}{ccccc}0& 0& 0& 0& 0\\ 0& 0& 1/2& 0& 0\\ 0& 1/2& 0& 1/2& 1\\ 0& 0& 1/2& 0& 0\\ 0& 0& 1& 0& 0\end{array}$ | ${M}^{2}(G,{F}_{2})=\begin{array}{ccccc}0& 0& 1& 0& 0\\ 0& 0& 0& 0& 1/2\\ 1& 0& 3& 0& 0\\ 0& 0& 0& 0& 1/2\\ 0& 1/2& 0& 1/2& 1\end{array}$ | |||

${M}^{0}(G,{F}_{3})=\begin{array}{ccccc}1& 0& 0& 0& 0\\ 0& 1& 0& 0& 0\\ 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0\end{array}$ | ${M}^{1}(G,{F}_{3})=\begin{array}{ccccc}0& 2& 0& 0& 0\\ 2& 0& 1/2& 0& 0\\ 0& 1/2& 0& 0& 0\\ 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0\end{array}$ | ${M}^{2}(G,{F}_{3})=\begin{array}{ccccc}4& 0& 1& 0& 0\\ 0& 5& 0& 1/2& 1/2\\ 1& 0& 0& 0& 0\\ 0& 1/2& 0& 0& 0\\ 0& 1/2& 0& 0& 0\end{array}$ |

#### The **TOMO-COMD** software

**TOMO-COMD**software [36]. This software has a graphical interface that makes it user friendly for medicinal chemists. The input of the chemical structure is by directly drawing the molecular pseudograph using the software’s drawing mode. This procedure is carried out by a selection of the active atom symbols belonging to different groups of the periodic table. The multiple edges and loops are edited with a right mouse click. Afterwards, in the calculation mode, one should select the atomic property and the family descriptor before calculating the molecular indices. In this work, we used the Mulliken electronegativity as an example of an atomic property [37]. The descriptors calculated were the following:

- (1)
**q**_{k}(x) and**q**_{k}(x) are the k-th total quadratic indices calculated using the k-th power of the matrices [^{H}**M**(G) or^{k}**M**(G^{k}^{H})] of the molecular pseudograph (G) considering and not considering hydrogen atoms, respectively.- (2)
^{E}**q**_{k}_{L}(x) [or^{E}**q**_{k}_{L}^{H}(x)] and^{H}**q**_{k}_{L}(x) are the k-th local quadratic indices calculated using a k-th power of the local matrices [**M**^{k}_{L}(G, F_{R})] of the molecular pseudograph (G) not considering (or considering) hydrogen atoms for heteroatoms (S,N,O) and hydrogen bonding heteroatoms, respectively.

#### Physical properties data sets for QSPR studies

- a)
- b)
- c)
- d)

**Table 4.**Quadratic Indices of the “Molecular Pseudograph’s Atom Adjacency Matrix” for C3-C9 Alkanes.

no. | Alkane | q_{0}^{H}(x) | q_{1}^{H}(x) | q_{2}^{H}(x) | q_{3}^{H}(x) | q_{4}^{H}(x) | q_{0}(x) | q_{2}(x) | q_{3}(x) | q_{5}(x) |
---|---|---|---|---|---|---|---|---|---|---|

1 | 2 | 42.8738 | 83.2658 | 211.8872 | 461.6846 | 1097.3462 | 13.8338 | 13.8338 | 13.8338 | 13.8338 |

2 | 3 | 59.4707 | 120.2436 | 319.0366 | 749.5692 | 1876.432 | 20.7507 | 41.5014 | 55.3352 | 110.6704 |

3 | 4 | 76.0676 | 157.2214 | 426.186 | 1037.8236 | 2666.8698 | 27.6676 | 69.169 | 110.6704 | 290.5098 |

4 | 2M3 | 76.0676 | 157.2214 | 426.5558 | 1048.8058 | 2757.6878 | 27.6676 | 83.0028 | 124.5042 | 373.5126 |

5 | 5 | 92.6645 | 194.1992 | 533.3354 | 1326.078 | 3457.6774 | 34.5845 | 96.8366 | 166.0056 | 498.0168 |

6 | 2M4 | 92.6645 | 194.1992 | 533.7052 | 1337.43 | 3559.4776 | 34.5845 | 110.6704 | 193.6732 | 664.0224 |

7 | 22MM3 | 92.6645 | 194.1992 | 534.4448 | 1359.3944 | 3741.1136 | 34.5845 | 138.338 | 221.3408 | 885.3632 |

8 | 6 | 109.2614 | 231.177 | 640.4848 | 1614.3324 | 4248.485 | 41.5014 | 124.5042 | 221.3408 | 719.3576 |

9 | 2M5 | 109.2614 | 231.177 | 640.8546 | 1625.6844 | 4350.655 | 41.5014 | 138.338 | 249.0084 | 899.197 |

10 | 3M5 | 109.2614 | 231.177 | 640.8546 | 1626.0542 | 4361.6372 | 41.5014 | 138.338 | 262.8422 | 982.1998 |

11 | 22MM4 | 109.2614 | 231.177 | 641.5942 | 1648.3884 | 4554.2554 | 41.5014 | 166.0056 | 304.3436 | 1314.211 |

12 | 23MM4 | 109.2614 | 231.177 | 641.2244 | 1637.4062 | 4463.4374 | 41.5014 | 152.1718 | 290.5098 | 1175.873 |

13 | 7 | 125.8583 | 268.1548 | 747.6342 | 1902.5868 | 5039.2926 | 48.4183 | 152.1718 | 276.676 | 940.6984 |

14 | 2M6 | 125.8583 | 268.1548 | 748.004 | 1913.9388 | 5141.4626 | 48.4183 | 166.0056 | 304.3436 | 1134.3716 |

15 | 3M6 | 125.8583 | 268.1548 | 748.004 | 1914.3086 | 5152.8146 | 48.4183 | 166.0056 | 318.1774 | 1231.2082 |

16 | 3E.5 | 125.8583 | 268.1548 | 748.004 | 1914.6784 | 5164.1666 | 48.4183 | 166.0056 | 332.0112 | 1328.0448 |

17 | 22MM5 | 125.8583 | 268.1548 | 748.7436 | 1936.6428 | 5345.8026 | 48.4183 | 193.6732 | 359.6788 | 1577.0532 |

18 | 23MM5 | 125.8583 | 268.1548 | 748.3738 | 1926.0304 | 5265.9668 | 48.4183 | 179.8394 | 359.6788 | 1521.718 |

19 | 24MM5 | 125.8583 | 268.1548 | 748.3738 | 1925.2908 | 5244.0024 | 48.4183 | 179.8394 | 332.0112 | 1328.0448 |

20 | 33MM5 | 125.8583 | 268.1548 | 748.7436 | 1937.3824 | 5367.767 | 48.4183 | 193.6732 | 387.3464 | 1770.7264 |

21 | 223MMM4 | 125.8583 | 268.1548 | 749.1134 | 1948.7344 | 5469.5672 | 48.4183 | 207.507 | 415.014 | 1992.0672 |

22 | 8 | 142.4552 | 305.1326 | 854.7836 | 2190.8412 | 5830.1002 | 55.3352 | 179.8394 | 332.0112 | 1162.0392 |

23 | 2M7 | 142.4552 | 305.1326 | 855.1534 | 2202.1932 | 5932.2702 | 55.3352 | 193.6732 | 359.6788 | 1355.7124 |

24 | 3M7 | 142.4552 | 305.1326 | 855.1534 | 2202.563 | 5943.6222 | 55.3352 | 193.6732 | 373.5126 | 1466.3828 |

25 | 4M7 | 142.4552 | 305.1326 | 855.1534 | 2202.563 | 5943.992 | 55.3352 | 193.6732 | 373.5126 | 1480.2166 |

26 | 3E.6 | 142.4552 | 305.1326 | 855.1534 | 2202.9328 | 5955.344 | 55.3352 | 193.6732 | 387.3464 | 1590.887 |

27 | 22MM6 | 142.4552 | 305.1326 | 855.893 | 2224.8972 | 6136.6102 | 55.3352 | 221.3408 | 415.014 | 1826.0616 |

28 | 23MM6 | 142.4552 | 305.1326 | 855.5232 | 2214.2848 | 6057.1442 | 55.3352 | 207.507 | 415.014 | 1784.5602 |

29 | 24MM6 | 142.4552 | 305.1326 | 855.5232 | 2213.915 | 6046.162 | 55.3352 | 207.507 | 401.1802 | 1673.8898 |

30 | 25MM6 | 142.4552 | 305.1326 | 855.5232 | 2213.5452 | 6034.4402 | 55.3352 | 207.507 | 387.3464 | 1563.2194 |

31 | 33MM6 | 142.4552 | 305.1326 | 855.893 | 2225.6368 | 6159.3142 | 55.3352 | 221.3408 | 442.6816 | 2047.4024 |

32 | 34MM6 | 142.4552 | 305.1326 | 855.5232 | 2214.6546 | 6068.4962 | 55.3352 | 207.507 | 428.8478 | 1881.3968 |

33 | 23ME5 | 142.4552 | 305.1326 | 855.5232 | 2214.6546 | 6068.866 | 55.3352 | 207.507 | 428.8478 | 1895.2306 |

34 | 33ME5 | 142.4552 | 305.1326 | 855.893 | 2226.3764 | 6181.6484 | 55.3352 | 221.3408 | 470.3492 | 2254.9094 |

35 | 223MMM5 | 142.4552 | 305.1326 | 856.2628 | 2237.3586 | 6272.4664 | 55.3352 | 235.1746 | 484.183 | 2365.5798 |

36 | 224MMM5 | 142.4552 | 305.1326 | 856.2628 | 2236.2492 | 6239.5198 | 55.3352 | 235.1746 | 442.6816 | 2033.5686 |

37 | 233MMM5 | 142.4552 | 305.1326 | 856.2628 | 2237.7284 | 6283.4486 | 55.3352 | 235.1746 | 498.0168 | 2476.2502 |

38 | 234MMM5 | 142.4552 | 305.1326 | 855.893 | 2226.0066 | 6170.6662 | 55.3352 | 221.3408 | 456.5154 | 2088.9038 |

39 | 2233MMMM4 | 147.2952 | 305.1326 | 857.0024 | 2260.4324 | 6487.049 | 55.3352 | 262.8422 | 553.352 | 3001.9346 |

40 | 9 | 159.0521 | 342.1104 | 961.933 | 2479.0956 | 6620.9078 | 62.2521 | 207.507 | 387.3464 | 1383.38 |

41 | 2M8 | 159.0521 | 342.1104 | 962.3028 | 2490.4476 | 6723.0778 | 62.2521 | 221.3408 | 415.014 | 1577.0532 |

42 | 3M8 | 159.0521 | 342.1104 | 962.3028 | 2490.8174 | 6734.4298 | 62.2521 | 221.3408 | 428.8478 | 1687.7236 |

43 | 4M8 | 159.0521 | 342.1104 | 962.3028 | 2490.8174 | 6734.7996 | 62.2521 | 221.3408 | 428.8478 | 1715.3912 |

44 | 3E.7 | 159.0521 | 342.1104 | 962.3028 | 2491.1872 | 6746.1516 | 62.2521 | 221.3408 | 442.6816 | 1826.0616 |

45 | 4E.7 | 159.0521 | 342.1104 | 962.3028 | 2491.1872 | 6746.5214 | 62.2521 | 221.3408 | 442.6816 | 1853.7292 |

46 | 22MM7 | 159.0521 | 342.1104 | 963.0424 | 2513.1516 | 6927.4178 | 62.2521 | 249.0084 | 470.3492 | 2047.4024 |

47 | 23MM7 | 159.0521 | 342.1104 | 962.6726 | 2502.5392 | 6847.9518 | 62.2521 | 235.1746 | 470.3492 | 2019.7348 |

48 | 24MM7 | 159.0521 | 342.1104 | 962.6726 | 2502.1694 | 6837.3394 | 62.2521 | 235.1746 | 456.5154 | 1922.8982 |

49 | 25MM7 | 159.0521 | 342.1104 | 962.6726 | 2502.1694 | 6836.5998 | 62.2521 | 235.1746 | 456.5154 | 1895.2306 |

50 | 26MM7 | 159.0521 | 342.1104 | 962.6726 | 2501.7996 | 6825.2478 | 62.2521 | 235.1746 | 442.6816 | 1770.7264 |

51 | 33MM7 | 159.0521 | 342.1104 | 963.0424 | 2513.8912 | 6950.1218 | 62.2521 | 249.0084 | 498.0168 | 2296.4108 |

52 | 34MM7 | 159.0521 | 342.1104 | 962.6726 | 2502.909 | 6859.6736 | 62.2521 | 235.1746 | 484.183 | 2144.239 |

53 | 35MM7 | 159.0521 | 342.1104 | 962.6726 | 2502.5392 | 6848.3216 | 62.2521 | 235.1746 | 470.3492 | 2019.7348 |

54 | 44MM7 | 159.0521 | 342.1104 | 963.0424 | 2513.8912 | 6950.8614 | 62.2521 | 249.0084 | 498.0168 | 2324.0784 |

55 | 23ME6 | 159.0521 | 342.1104 | 962.6726 | 2502.909 | 6860.0434 | 62.2521 | 235.1746 | 484.183 | 2171.9066 |

56 | 24ME6 | 159.0521 | 342.1104 | 962.6726 | 2502.5392 | 6848.6914 | 62.2521 | 235.1746 | 470.3492 | 2047.4024 |

57 | 33ME6 | 159.0521 | 342.1104 | 963.0424 | 2514.6308 | 6973.1956 | 62.2521 | 249.0084 | 525.6844 | 2545.4192 |

58 | 34ME6 | 159.0521 | 342.1104 | 962.6726 | 2503.2788 | 6871.3954 | 62.2521 | 235.1746 | 498.0168 | 2268.7432 |

59 | 223MMM6 | 159.0521 | 342.1104 | 963.4122 | 2525.613 | 7063.6438 | 62.2521 | 262.8422 | 539.5182 | 2642.2558 |

60 | 224MMM6 | 159.0521 | 342.1104 | 963.4122 | 2524.8734 | 7041.6794 | 62.2521 | 262.8422 | 511.8506 | 2393.2474 |

61 | 225MMM6 | 159.0521 | 342.1104 | 963.4122 | 2524.5036 | 7029.5878 | 62.2521 | 262.8422 | 498.0168 | 2268.7432 |

62 | 233MMM6 | 159.0521 | 342.1104 | 963.4122 | 2525.9828 | 7074.9958 | 62.2521 | 262.8422 | 553.352 | 2766.76 |

63 | 234MMM6 | 159.0521 | 342.1104 | 963.0424 | 2514.6308 | 6973.1956 | 62.2521 | 249.0084 | 525.6844 | 2462.4164 |

64 | 235MMM6 | 159.0521 | 342.1104 | 963.0424 | 2513.8912 | 6950.4916 | 62.2521 | 249.0084 | 498.0168 | 2241.0756 |

65 | 244MMM6 | 159.0521 | 342.1104 | 963.4122 | 2525.2432 | 7053.0314 | 62.2521 | 262.8422 | 525.6844 | 2517.7516 |

66 | 334MMM6 | 159.0521 | 342.1104 | 963.4122 | 2526.3526 | 7086.3478 | 62.2521 | 262.8422 | 567.1858 | 2863.5966 |

67 | 33EE5 | 159.0521 | 342.1104 | 963.0424 | 2515.3704 | 6995.8996 | 62.2521 | 249.0084 | 553.352 | 2766.76 |

68 | 223MME5 | 159.0521 | 342.1104 | 963.4122 | 2525.9828 | 7075.7354 | 62.2521 | 262.8422 | 553.352 | 2766.76 |

69 | 233MME5 | 159.0521 | 342.1104 | 963.4122 | 2526.7224 | 7097.6998 | 62.2521 | 262.8422 | 581.0196 | 2988.1008 |

70 | 234MEM5 | 159.0521 | 342.1104 | 963.0424 | 2514.6308 | 6973.9352 | 62.2521 | 249.0084 | 525.6844 | 2490.084 |

71 | 2233(M)5 | 159.0521 | 342.1104 | 964.1518 | 2549.4264 | 7301.3002 | 62.2521 | 290.5098 | 636.3548 | 3513.7852 |

72 | 2234(M)5 | 159.0521 | 342.1104 | 963.782 | 2537.3348 | 7177.5356 | 62.2521 | 276.676 | 581.0196 | 2960.4332 |

73 | 2244(M)5 | 159.0521 | 342.1104 | 964.1518 | 2547.2076 | 7235.407 | 62.2521 | 290.5098 | 553.352 | 2766.76 |

74 | 2334(M)5 | 159.0521 | 342.1104 | 963.782 | 2538.0744 | 7199.5 | 62.2521 | 276.676 | 608.6872 | 3209.4416 |

no. | q_{7}(x) | q_{11}(x) | q_{13}(x) | q_{15}(x) | no. | q_{7}(x) | q_{11}(x) | q_{13}(x) | q_{15}(x) | |

1 | 13.8338 | 13.8338 | 13.8338 | 13.8338 | 38 | 9531.4882 | 198335.19 | 904716.69 | 4126913 | |

2 | 221.3408 | 885.3632 | 1770.7264 | 3541.4528 | 39 | 16033.374 | 452213.09 | 2398545.7 | 12719902 | |

3 | 760.859 | 5215.3426 | 13653.9606 | 35746.539 | 40 | 4980.168 | 65018.86 | 235174.6 | 850778.7 | |

4 | 1120.5378 | 10084.84 | 30254.5206 | 90763.562 | 41 | 6031.5368 | 88812.996 | 341252.18 | 1311776.3 | |

5 | 1494.0504 | 13446.454 | 40339.3608 | 121018.08 | 42 | 6695.5592 | 106326.59 | 424531.65 | 1696120.7 | |

6 | 2268.7432 | 26450.226 | 90307.0464 | 308327.73 | 43 | 6930.7338 | 114032.01 | 463003.45 | 1880234.8 | |

7 | 3541.4528 | 56663.245 | 226652.979 | 906611.92 | 44 | 7580.9224 | 131365.76 | 547431.13 | 2281968.3 | |

8 | 2337.9122 | 24665.665 | 80097.702 | 260089.27 | 45 | 7802.2632 | 138504.01 | 583675.69 | 2459760.3 | |

9 | 3250.943 | 42538.935 | 153901.025 | 556810.45 | 46 | 9019.6376 | 178179.34 | 795498.84 | 3556836 | |

10 | 3665.957 | 51060.556 | 190560.595 | 711181.82 | 47 | 8715.294 | 163432.51 | 708926.91 | 3076720.1 | |

11 | 5658.0242 | 104763.37 | 450774.373 | 1939581.8 | 48 | 8148.1082 | 146859.62 | 623696.87 | 2648840.7 | |

12 | 4717.3258 | 75546.382 | 302199.361 | 1208811.3 | 49 | 7871.4322 | 135571.24 | 562274.8 | 2331382.6 | |

13 | 3209.4416 | 37406.595 | 127713.642 | 436041.38 | 50 | 7082.9056 | 113326.49 | 453305.96 | 1813223.8 | |

14 | 4233.1428 | 58959.656 | 220040.423 | 821202.04 | 51 | 10679.694 | 233182.53 | 1091597.5 | 5112419.1 | |

15 | 4772.661 | 71797.422 | 278515.895 | 1080447.4 | 52 | 9545.322 | 189772.07 | 846421.05 | 3775354.7 | |

16 | 5312.1792 | 84994.867 | 339979.469 | 1359917.9 | 53 | 8687.6264 | 160831.76 | 692022.01 | 2977614.8 | |

17 | 6944.5676 | 135156.23 | 596541.124 | 2633153.2 | 54 | 10956.37 | 245024.27 | 1159383.1 | 5486153.1 | |

18 | 6418.8832 | 114045.85 | 480641.547 | 2025600.3 | 55 | 9752.829 | 196688.97 | 883301.96 | 3966786.8 | |

19 | 5312.1792 | 84994.867 | 339979.469 | 1359917.9 | 56 | 8908.9672 | 168329.68 | 731310 | 3176683.2 | |

20 | 8078.9392 | 168108.34 | 766835.202 | 3497959.3 | 57 | 12367.417 | 292640.21 | 1423940.7 | 6928990.7 | |

21 | 9462.3192 | 212155.16 | 1004001.87 | 4751080.3 | 58 | 10347.682 | 215309.26 | 982144.46 | 4480103.8 | |

22 | 4094.8048 | 51060.556 | 180365.084 | 637115.66 | 59 | 12962.271 | 312505.54 | 1534901.6 | 7539338 | |

23 | 5132.3398 | 73886.326 | 280632.467 | 1066267.8 | 60 | 11219.212 | 247237.67 | 1161361.3 | 5456410.4 | |

24 | 5782.5284 | 90279.379 | 356995.043 | 1411988.3 | 61 | 10347.682 | 215309.26 | 982144.46 | 4480103.8 | |

25 | 5907.0326 | 94443.353 | 377759.577 | 1511024.5 | 62 | 13833.8 | 345845 | 1729225 | 8646125 | |

26 | 6543.3874 | 110767.24 | 455782.209 | 1875475.9 | 63 | 11537.389 | 253324.55 | 1186995.4 | 5561796.3 | |

27 | 8092.773 | 160236.91 | 714156.091 | 3184194.9 | 64 | 10071.006 | 202803.51 | 909378.68 | 4076654.9 | |

28 | 7677.759 | 142142.3 | 611606.132 | 2631603.8 | 65 | 12090.741 | 279636.43 | 1345613.7 | 6476127.5 | |

29 | 6986.069 | 121585.27 | 507105.607 | 2114869.8 | 66 | 14456.321 | 368504.76 | 1860549.3 | 9393758.9 | |

30 | 6294.379 | 101443.26 | 406533.881 | 1628127.6 | 67 | 13833.8 | 345845 | 1729225 | 8646125 | |

31 | 9503.8206 | 205390.43 | 955196.222 | 4442545 | 68 | 13833.8 | 345845 | 1729225 | 8646125 | |

32 | 8258.7786 | 159185.54 | 698869.742 | 3068226.2 | 69 | 15327.85 | 402729.59 | 2064003 | 10577877 | |

33 | 8369.449 | 163114.34 | 720035.456 | 3178412.4 | 70 | 11786.398 | 263948.9 | 1249026.1 | 5910463.4 | |

34 | 10804.198 | 248026.2 | 1188364.92 | 5693798.4 | 71 | 19228.982 | 572193.64 | 3118774.9 | 16996954 | |

35 | 11523.555 | 273010.04 | 1328584.32 | 6465267.9 | 72 | 15051.174 | 389172.46 | 1979229.4 | 10066248 | |

36 | 9365.4826 | 199303.56 | 920044.537 | 4247972.6 | 73 | 13833.8 | 345845 | 1729225 | 8646125 | |

37 | 12242.913 | 298408.9 | 1472843.18 | 7269219.2 | 74 | 16821.901 | 461274.23 | 2415270.8 | 12646528 |

#### Data analysis

^{2}) have been used as a means of indicating predictive ability. Many authors consider high q

^{2}values (for instance, q

^{2}> 0.5) as indicator or even as the ultimate proof of the high predictive power of a QSAR model. In a recent paper, Golbraikh and Tropsha demonstrated that high values of LOO q

^{2}appears to be a necessary but not the sufficient condition for the model to have a high predictive power [48]. A more exhaustive cross-validation method can be used in which a fraction of the data (10-20%) is left out and predicted from a model based on the remaining data. This process (leave-group-out, LGO) is repeated until each observation has been left out at least once [49,50]. For this present paper, each investigated data set was splited randomly into five groups of approximately the same size (20%). Each group was left out (LGO) and that group was then predicted by a model developed from the remaining observations (80% of the data). This process was carried out five times on five unique subsets. In this way, every observation was left out once, in groups of 20%, and its value predicted. The mean absolute errors (MAE) for the five groups will be used as the significant criterion for assessing model quality. The level of overall (average) MAE (for a 20% full leave-out) of 5-fold cross-validation procedure can be taken as good confirmation of the predictive quality of the model. In addition, to assess the robustness and predictive power of the found models, external prediction (test) sets also were used. This type of model validation is very important, if we take into consideration that the predictive ability of a QSAR model can only be estimated using an external test set of compounds that was not used for building the model [48].

#### QSPR applications

**P**and the quadratic indices of

**M**having, for instance, the following appearance:

**P**=a

_{0}

**q**

_{0}(x) + a

_{1}

**q**

_{1}(x) + a

_{2}

**q**

_{2}(x) +….+ a

_{k}

**q**

_{k}(x) + c

**P**is the measurement of the property,

**q**

_{k}(x) [or

**q**

_{kL}(x)] is the kth total [or local] quadratic indices, and the

**a**

_{k}’s are the coefficients obtained by the linear regression analysis.

**R**is the multiple correlation coefficient,

**s**is the standard deviation of the regression,

**q**is the square multiple correlation coefficient of the LOO cross-validation procedure;

^{2}**MAE**is the (average) mean absolute error of the LGO cross-validation procedure;

**F**is the Fisher ratio at the 95% confidence level, and the

**p**-value is the significance level.

**Table 5.**Multiple Regression Equation for Physical Properties Using the Quadratic Indices of the Molecular Pseudograph’s Atom Adjacency Matrix.

B.p. (^{o}C)=-204.184(±3.262) +1.44048(±0.026)^{.}q(x) -9.29x10_{1}^{H}^{-3}(±0.427x10^{-3})^{.}q(x)_{0}^{.}q(x) +2.91x10_{2}^{-7} (±1.75x10^{-8})^{.}q(x)_{0}^{.}q(x) -0.11678(±0.028)_{13}^{.}q(x)
_{2}N=74 R=0.9988 q ^{2}=0.9970 F(4.69)=7068.1 s=2.35 MAE=2.11 p<0.0000MV (cm^{3})=39.72(±2.441) +0.7651(±0.031)^{.}q_{0}(x) -4.4x10^{H}^{-7}(±1.08x10^{-7})^{.}q(x) +4.634x10_{15}^{-3}(±0.214 x10^{-3})^{.}q(x)_{0}(x) -1.74x10^{.}q_{2}^{-3}(±0.132x10^{-3})^{.}q(x)_{0}^{.}q(x)
_{3}N=69 R=0.9991 q ^{2}=0.9973 F(4.69)=8916.5 s= 0.75 MAE=0.53 p<0.0000MR (cm^{3})=3.2327(±0.048) +1.734x10^{-2}(±4.71x10^{-5})^{.} q(x) _{3}^{H}-0.01012(±0.302x10^{-3})^{.}q(x) +7.486x10_{3}^{-3} (±0.836x10^{-3})^{.}q(x)
_{2}N= 69 R=0.9999 q ^{2}=0.9999 F(3.65)= 2.52x10^{5} s= 0.049 MAE=0.0322 p<0.00HV (KJ/mol)=-1.35607(±0.327) +0.07648(±0.001)^{.}q(x) -0.1309(±0.004)_{2}^{H}^{.}q(x) +1.19x10_{2}^{-5}(±9.3x10^{-7}) ^{.}q(x)
_{11}N=69 R=0.998 q ^{2}= 0.9955 F(3.65)=5469.5 s= 0.34 MAE=0.32 p<0.0000TC (^{o}C)=-71.6809(±6.373) +0.2399(±0.007)^{.}q(x) -0.02165(±0.001)_{3}^{H}^{.}q(x)_{0}^{.}q(x) +0.83x10_{2}^{-3}(±6.01x10^{-5}) ^{.}q(x)_{0}^{.} q(x)
_{5}N=74 R=0.9953 q ^{2}= 0.9892 F(3.70)=2460.1 s=5.66 MAE=5.34 p<0.0000PC (atm)=54.7074(±0.786) -6.998x10^{-3}(±0.265x10^{-3})^{.}q(x)+5.95x10_{4}^{H}^{-4}(±3.72x10^{-5})^{.}q(x)_{0}^{.}q(x)
_{3}N=74 R=0.9803 q ^{2}=0.9575 F(2.71)= 878.64 s= 0.86 MAE=0.64 p<0.0000ST (dyn/cm)=-3.49402(±1.097) +0.04848(±0.001)^{.}q(x)_{2}^{H}-0.00163(±0.122x10 ^{-3})^{.}q(x)_{0}^{.}q(x) +1.21x10_{2}^{-5}(±5.15x10^{-7})^{.}q(x)_{0}^{.}q(x)_{7}-0.01617(±0.006) ^{.}q(x)
_{2}N=68 R=0.9892 q ^{2}= 0.9734 F(4.63)=722.14 s= 0.29 MAE=0.23 p<0.0000 |

**Table 6.**Statistical Parameters for the Models Describing Physical Properties of Alkanes by Using Conectivity Indices, ad hoc Descriptors, Spectral Moments of Edge-Adjacency Matrix and Quadratic Indices of the Molecular Pseudograph’s Atom.Adjacency Matrix.

Connectivity Indices | ad hoc Descriptors | Moments of E Matrix | Quadratic Indices of M Matrix | |||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|

Prop. | n^{a} | R | s | n^{a} | R | s | n^{a} | R | s | n^{a} | R | s |

Bp | 5 | 0.9995 | 1.86 | 5 | 0.9989 | 2.0 | 4 | 0.9984 | 2.48 | 4 | 0.9988 | 2.35 |

MV | 5 | 0.9995 | 0.5 | 5 | 0.9995 | 0.4 | 5 | 0.9993 | 0.6 | 4 | 0.9991 | 0.75 |

MR | 5 | 0.9999 | 0.05 | 5 | 0.9999 | 0.05 | 4 | 0.9999 | 0.05 | 3 | 0.9999 | 0.05 |

HV | 5 | 0.9989 | 0.2 | 5 | 0.9969 | 0.4 | 3 | 0.9988 | 0.2 | 3 | 0.9980 | 0.34 |

TC | 5 | 0.9975 | 4.1 | 5 | 0.9970 | 4.8 | 5 | 0.9944 | 5.4 | 3 | 0.9953 | 5.66 |

PC | 5 | 0.9904 | 0.6 | 5 | 0.9889 | 0.7 | 5 | 0.9854 | 0.6 | 2 | 0.9803 | 0.86 |

ST | 5 | 0.9929 | 0.2 | 5 | 0.9945 | 0.2 | 6 | 0.9869 | 0.3 | 4 | 0.9892 | 0.29 |

^{a}Number of Variables in QSPR Models.

**.**[44] from which the Bp of the 58 alkyl alcohols have been computed, which have been used in several QSAR/QSAR studies [52,53,54,55,56].

**Bp (**=34.16625(±2.696) +0.26497(±0.0111)

^{o}C)^{.}

**q**(x) -0.29237(±0.045)

_{2}^{H}^{.}

**q**(x)

_{2}-78.0818x10

^{–5}(±9.932x10

^{-5})

^{.E}

**q**(x)

_{9L}^{H}**Bp**

**(**=461.7348(±30.20806) +0.092098(±0.002)

^{o}C)^{.}

**q**(x) -0.0175226(±0.001)

_{3}^{H}^{.}

**q**(x)

_{6}-10.266162(±0.707)

^{.E}

**q**(x) +10.956280x10

_{2L}^{H}^{-5}(±1.32x10

^{-5})

^{.E }

**q**

**(x)**

_{14L}^{2}) for equations 26 and 27 were 0.9877 and 0.9977, respectively. Therefore, these models explained more than 98% and 99% of the variance for the experimental values of Bp [57,58].

Equation | Set | Correlation Coefficient (R) | Standard Error (S) | Fischer ratio (F) | Average Deviation |
---|---|---|---|---|---|

Eq. 26 | Complete | 0.9938 | 4.006 | 1446.9 | 2.82 |

Randić and Basak /48/ | Complete | 0.9938 | 4.039 | 2193 | 2.90 |

Eq. 27 | Training Test | 0.9979 0.9938 | 2.97 3.17 | 1390.7 2177.9 | 2.13 2.15 |

Eq. 11 /44/ | Training Test | 0.9953 0.9948 | 2.903 3.025 | 5733 2529 | 2.20 2.50 |

Eq. 12 /44/ | Training Test | 0.9953 0.9948 | 3.008 2.833 | 2764 1296 | 2.20 2.48 |

Eq. 13 /44/ | Training Test | 0.9954 0.9949 | 2.874 2.871 | 2018 841 | 2.03 2.63 |

^{2}) of 0.986 and 0.992, respectively.

^{o}C (MAE=1.579, 1.728, 2.674, 3.546 and 3.375

^{o}C). The overall MAE were 3.824

^{o}C and 2.580

^{o}C for the models 26 and 27, respectively. For a 20% full leave-out cross-validation procedure, this level of MAE is good confirmation of the predictive quality of the models developed.

**s**) and the average of the deviation obtained by us are smaller.

Alkyl alcohol | Bp exp (^{o}C) | Bp calc. (Eq.26) | ∆^{*} | % ∆ | Bp cal. Ref./48/ |
---|---|---|---|---|---|

1. methanol | 64.70 | 65.50 | -0.80 | -1.24 | 65.24 (-0.54) |

2. ethanol | 78.30 | 78.43 | -0.13 | -0.17 | 77.69 (0.61) |

3. 1-propanol | 97.20 | 95.63 | 1.57 | 1.62 | 96.42 (0.77) |

4. 2. propanol | 82.30 | 85.83 | -3.53 | -4.28 | 84.11 (-1.81) |

5. 1-butanol | 117.70 | 113.40 | 4.30 | 3.65 | 115.67 (2.03) |

6. 2-butanol | 99.60 | 102.87 | -3.27 | -3.28 | 102.43 (-2.83) |

7. 2-methyl-1-propanol | 107.90 | 108.66 | -0.76 | -0.71 | 109.15 (-1.25) |

8. 2-methyl-2-propanol | 82.40 | 87.68 | -5.28 | -6.41 | 84.52 (-2.12) |

9. 1-pentanol | 137.80 | 133.16 | 4.64 | 3.36 | 134.92 (2.88) |

10. 2-pentanol | 119.00 | 120.59 | -1.59 | -1.34 | 121.68 (-2.68) |

11. 3-pentanol | 115.30 | 119.90 | -4.60 | -3.99 | 120.75 (-5.45) |

12. 2-methyl-1-butanol | 128.70 | 126.39 | 2.31 | 1.80 | 127.97 (0.73) |

13. 3-methyl-1-butanol | 131.20 | 127.13 | 4.07 | 3.10 | 128.90 (2.30) |

14. 2.methyl-2-butanol | 102.00 | 104.57 | -2.57 | -2.52 | 102.41 (-0.41) |

15. 3-methyl-2-butanol | 111.50 | 115.75 | -4.25 | -3.81 | 114.72 (-3.22) |

16. 2,2-dimethyl-1-propanol | 113.10 | 117.54 | -4.44 | -3.93 | 115.84 (-2.74) |

17. 1-hexanol | 157.13 | 153.12 | 4.01 | 2.55 | 154.17 (2.83) |

18. 2-hexanol | 139.90 | 140.35 | -0.45 | -0.32 | 140.92 (-1.02) |

19. 3-hexanol | 135.40 | 137.63 | -2.23 | -1.64 | 139.99 (-4.59) |

20. 2-methyl-1-pentanol | 148.00 | 146.14 | 1.86 | 1.25 | 147.22 (0.78) |

21.3-methyl-1-pentanol | 152.40 | 146.89 | 5.51 | 3.61 | 147.72 (4.8) |

22. 4-methyl-1-pentanol | 151.80 | 148.97 | 2.83 | 1.86 | 148.15 (3.65) |

23. 2-methyl-2-pentanol | 121.40 | 122.25 | -0.85 | -0.70 | 121.66 (-0.25) |

24. 3-methyl-2-pentanol | 134.20 | 133.42 | 0.78 | 0.58 | 133.55 (0.65) |

25. 4-methyl-2-pentanol | 131.70 | 134.27 | -2.57 | -1.95 | 134.90 (-3.20) |

26. 2-methyl-3-pentanol | 126.50 | 132.77 | -6.27 | -4.96 | 134.31 (-7.81) |

27. 3-methyl-3-pentanol | 122.40 | 121.45 | 0.95 | 0.78 | 120.30 (2.10) |

28. 2-ethyl-1-butanol | 146.50 | 144.11 | 2.39 | 1.63 | 146.79 (-0.29) |

29. 2,2-dimethyl-1-butanol | 136.80 | 135.21 | 1.59 | 1.16 | 134.37 (2.43) |

30. 2,3-dimethyl-1-butanol | 149.00 | 140.07 | 8.93 | 6.00 | 140.77 (8.23) |

31. 3.3-dimethyl-1-butanol | 143.00 | 136.82 | 6.18 | 4.32 | 136.11 (6.89) |

32. 2,3-dimethyl-2-butanol | 118.60 | 117.30 | 1.30 | 1.10 | 114.28 (4.32) |

33. 3,3-dimethyl-2-butanol | 120.00 | 124.47 | -4.47 | -3.72 | 121.00 (-1.00) |

34. 1-heptanol | 176.30 | 173.38 | 2.92 | 1.66 | 173.41 (2.87) |

35. 3-heptanol | 156.80 | 157.38 | -0.58 | -0.37 | 159.24 (-2.44) |

36. 4-heptanol | 155.00 | 155.35 | -0.35 | -0.23 | 159.24 (-4.24) |

37. 2-methyl-2-hexanol | 142.50 | 142.00 | 0.50 | 0.35 | 140.90 (1.60) |

38. 3-methyl-3-hexanol | 142.40 | 139.13 | 3.27 | 2.30 | 139.55 (2.85) |

39. 3-ethyl-3-pentanol | 142.50 | 138.32 | 4.18 | 2.93 | 138.37 (4.13) |

40. 2,3-dimethyl-2-pentanol | 139.70 | 134.92 | 4.78 | 3.42 | 133.11 (6.59) |

41.3,3-dimethyl-2-pentanol | 133.00 | 142.09 | -9.09 | -6.83 | 139.67 (-6.57) |

42. 2.2-dimethyl-3-pentanol | 136.00 | 141.49 | -5.49 | -4.04 | 139.32 (-3.32) |

43. 2,3-dimethyl-3-pentanol | 139.00 | 134.17 | 4.83 | 3.48 | 132.18 (6.82) |

44. 2,4-dimethyl-3-pentanol | 138.80 | 145.64 | -6.84 | -4.93 | 145.34 (-6.54) |

45. 1-octanol | 195.20 | 193.67 | 1.53 | 0.78 | 192.58 (2.62) |

46. 2-octanol | 179.80 | 180.57 | -0.77 | -0.43 | 179.33 (0.47) |

47. 2-ethyl-1-hexanol | 184.60 | 183.82 | 0.78 | 0.42 | 185.29 (-0.69) |

48. 2,2,3trimethyl-3-pentanol | 152.20 | 142.73 | 9.47 | 6.22 | 152.78 (-0.57) |

49. 1-nonanol | 213.10 | 213.97 | -0.87 | -0.41 | 211.91 (1.19) |

50. 2-nonanol | 198.50 | 200.85 | -2.35 | -1.19 | 198.66 (-0.16) |

51. 3-nonanol | 194.70 | 197.60 | -2.90 | -1.49 | 197.73 (-3.03) |

52. 4-nonanol | 193.00 | 195.07 | -2.07 | -1.07 | 197.73 (-4.73) |

53. 5-nonanol | 195.10 | 194.87 | 0.23 | 0.12 | 197.73 (-2.63) |

54. 7-methyl-1-octanol | 206.00 | 210.01 | -4.01 | -1.95 | 205.46 (0.54) |

55. 2,6-dimethyl-4-heptanol | 178.00 | 182.72 | -4.72 | -2.65 | 185.69 (-7.69) |

56. 3,5-dimethyl-4-hexanol | 187.00 | 180.99 | 6.01 | 3.21 | 183.83 (3.17) |

57. 3,3,5-trimethyl-1-hexanol | 193.00 | 192.54 | 0.46 | 0.24 | 186.98 (6.02) |

58. 1-decanol | 230.20 | 234.27 | -4.07 | -1.77 | 231.15 (-0.95) |

^{*}Residual, defined as [Bp exp.– Bp calc], given in brackets for Ref. /48/.

Alkyl alcohol | Bp exp (^{o}C) | Bp calc. (Eq. 27) | ∆* | % ∆ | Bp calc. (Eq. 11) |
---|---|---|---|---|---|

1. methanol | 64.70 | 66.03 | -1.33 | -2.06 | 64.68 (0.02) |

2. ethanol | 78.30 | 75.96 | 2.34 | 2.99 | 77.36 (0.94) |

3. 1-propanol | 97.20 | 97.44 | -0.24 | -0.24 | 96.80 (0.40) |

4. 2. propanol | 82.30 | 80.69 | 1.61 | 1.96 | 78.24 (4.06) |

6.2-butanol | 99.60 | 100.08 | -0.48 | -0.48 | 97.68 (1.92) |

8. 2-methyl-2-propanol | 82.40 | 81.63 | 0.77 | 0.93 | 84.97 (-2.57) |

9. 1-pentanol | 137.80 | 137.06 | 0.74 | 0.54 | 135.69 (2.11) |

11. 3-pentanol | 115.30 | 118.40 | -3.10 | -2.69 | 117.13 (-1.83) |

14. 2.methyl-2-butanol | 102.00 | 101.74 | 0.26 | 0.26 | 104.41 (-2.41) |

16. 2,2-dimethyl-1-propanol | 113.10 | 116.94 | -3.84 | -3.40 | 117.11 (4.01) |

18. 2-hexanol | 139.90 | 138.73 | 1.17 | 0.83 | 136.57 (3.33) |

20. 2-methyl-1-pentanol | 148.00 | 147.82 | 0.18 | 0.12 | 148.68 (-0.68) |

22. 4-methyl-1-pentanol | 151.80 | 149.11 | 2.69 | 1.77 | 148.68 (3.12) |

26. 2-methyl-3-pentanol | 126.50 | 131.41 | -4.91 | -3.88 | 130.11 (-3.61) |

27. 3-methyl-3-pentanol | 122.40 | 121.41 | 0.99 | 0.81 | 123.86 (-1.46) |

29. 2,2-dimethyl-1-butanol | 136.80 | 132.03 | 4.77 | 3.49 | 136.55 (0.25) |

34. 1-heptanol | 176.30 | 175.42 | 0.88 | 0.50 | 174.57 (1.73) |

35. 3-heptanol | 156.80 | 156.88 | -0.08 | -0.05 | 156.01 (0.79) |

37. 2-methyl-2-hexanol | 142.50 | 140.89 | 1.61 | 1.13 | 143.30 (-0.80) |

39. 3-ethyl-3-pentanol | 142.50 | 140.75 | 1.75 | 1.23 | 143.30 (-0.80) |

41.3,3-dimethyl-2-pentanol | 133.00 | 136.16 | -3.16 | -2.37 | 137.43 (-4.43) |

44. 2,4-dimethyl-3-pentanol | 138.80 | 143.48 | -4.68 | -3.37 | 143.10 (-4.30) |

45. 1-octanol | 195.20 | 194.46 | 0.74 | 0.38 | 194.01 (1.19) |

48. 2,2,3trimethyl-3-pentanol | 152.20 | 154.18 | -1.98 | -1.30 | 144.16 (8.04) |

49. 1-nonanol | 213.10 | 213.32 | -0.22 | -0.10 | 213.45 (-0.35) |

52. 4-nonanol | 193.00 | 195.49 | -2.49 | -1.29 | 194.89 (-1.89) |

53. 5-nonanol | 195.10 | 195.34 | -0.24 | -0.12 | 194.89 (0.21) |

56. 3,5-dimethyl-4-hexanol | 187.00 | 178.80 | 8.20 | 4.39 | 181.99 (5.01) |

58. 1-decanol | 230.20 | 232.18 | -1.98 | -0.86 | 232.86 (-2.66) |

^{*}Residual, defined as [Bp exp. – Bp calc] given in brackets for Eq. 11. Ref. [44].

Alkyl alcohol | Bp exp. (^{o}C) | Bp calc. (Eq. 27) | ∆^{*} | % ∆ | Bp calc.(Eq. 11) |
---|---|---|---|---|---|

5. 1-butanol | 117.70 | 117.50 | 0.20 | 0.17 | 116.25 (1.45) |

7. 2-methyl-1-propanol | 107.90 | 112.68 | -4.78 | -4.43 | 109.79 (-1.89) |

10. 2-pentanol | 119.00 | 119.23 | -0.23 | -0.20 | 117.13 (1.87) |

12. 2-methyl-1-butanol | 128.70 | 130.00 | -1.30 | -1.01 | 129.34 (-0.64) |

13. 3-methyl-1-butanol | 131.20 | 131.11 | 0.09 | 0.07 | 129.23 (1.97) |

15. 3-methyl-2-butanol | 111.50 | 114.17 | -2.67 | -2.39 | 110.67 (0.83) |

17. 1-hexanol | 157.13 | 156.38 | 0.75 | 0.48 | 155.13 (1.87) |

19. 3-hexanol | 135.40 | 137.52 | -2.12 | -1.57 | 136.57 (-1.17) |

21.3-methyl-1-pentanol | 152.40 | 147.35 | 5.05 | 3.31 | 148.68 (3.72) |

23. 2-methyl-2-pentanol | 121.40 | 121.16 | 0.24 | 0.20 | 123.86 (-2.46) |

24. 3-methyl-2-pentanol | 134.20 | 131.27 | 2.93 | 2.18 | 130.11 (4.09) |

25. 4-methyl-2-pentanol | 131.70 | 132.55 | -0.85 | -0.65 | 130.11 (1.59) |

28. 2-ethyl-1-butanol | 146.50 | 146.12 | 0.38 | 0.26 | 148.68 (-2.18) |

30. 2,3-dimethyl-1-butanol | 149.00 | 141.00 | 8.00 | 5.37 | 142.22 (6.78) |

31. 3.3-dimethyl-1-butanol | 143.00 | 133.59 | 9.41 | 6.58 | 136.55 (6.45) |

32. 2,3-dimethyl-2-butanol | 118.60 | 119.44 | -0.84 | -0.71 | 117.40 (1.20) |

33. 3,3-dimethyl-2-butanol | 120.00 | 120.08 | -0.08 | -0.06 | 117.99 (2.01) |

36. 4-heptanol | 155.00 | 156.58 | -1.58 | -1.02 | 156.01 (-1.01) |

38. 3-methyl-3-hexanol | 142.40 | 141.12 | 1.28 | 0.90 | 143.30 (-0.90) |

40. 2,3-dimethyl-2-pentanol | 139.70 | 138.02 | 1.68 | 1.20 | 136.84 (2.86) |

42. 2.2-dimethyl-3-pentanol | 136.00 | 136.45 | -0.45 | -0.33 | 137.43 (-1.43) |

43. 2,3-dimethyl-3-pentanol | 139.00 | 138.90 | 0.10 | 0.07 | 136.84 (2.16) |

46. 2-octanol | 179.80 | 177.28 | 2.52 | 1.40 | 175.45 (4.35) |

47. 2-ethyl-1-hexanol | 184.60 | 182.69 | 1.91 | 1.03 | 187.56 (-2.96) |

50. 2-nonanol | 198.50 | 196.41 | 2.09 | 1.05 | 194.89 (3.61) |

51. 3-nonanol | 194.70 | 195.53 | -0.83 | -0.43 | 194.89 (-0.19) |

54. 7-methyl-1-octanol | 206.00 | 205.50 | 0.50 | 0.24 | 207.00 (1.00) |

55. 2,6-dimethyl-4-heptanol | 178.00 | 183.63 | -5.63 | -3.16 | 181.99 (-3.99) |

57. 3,3,5-trimethyl-1-hexanol | 193.00 | 190.45 | 2.55 | 1.32 | 188.43 (4.57) |

^{*}Residual, defined as [Bp exp.– Bp calc], given in brackets for Eq. 11. Ref. [44].

**Bp (**=-105.146(±4.718) +3.1629(±0.118)

^{o}C)^{.}

**q**(x) -0.4933(±0.045)

_{1}^{.}

**q**(x)

_{2}**Bp**

**(**=-108.197(±3.635) +1.6358(±0.361)

^{o}C)^{.}

**q**(x) +2.038(±0.103)

_{0}^{.}

**q**(x)

_{1}-0.3016(±4.718)

^{.}

**q**(x) -1.75x10

_{2}^{-5}(±3.75x10

^{-6})

^{.}

**q**(x)

_{14}+6.42x10

^{-6}(±1.34x10

^{-6})

^{.}

**q**(x)

_{15}**Table 11.**Statistical Parameters Corresponding to the Regression Equations for 80 Compounds Present in the Training Data Set.

Equation | Set | Correlation Coefficient (R) | Standard Error (S) | Fischer ratio (F) |
---|---|---|---|---|

Eq. (28) two descriptors | Training Test | 0.9823 0.9726 | 7.8211 10.245 | 1058.2 421.21 |

Eq. (29) Five descriptors | Training Test | 0.9927 0.9938 | 5.0145 4.7865 | 5257.9 2025.4 |

Eq. (1)/(25). Six descriptors | Training Test | 0.9937 0.9943 | 4.800 4.696 | 960 2094.8 |

no | Cycloalkane | Obsd (^{o}C) | Cald [Eq. 28] | Res. | Cald [Eq. 29] | Res. | Cald [Eq. 1 /25 ] | Res. |
---|---|---|---|---|---|---|---|---|

1 | cyclopropane | -32.8 | -14.82 | -17.98 | -16.07 | -16.73 | -36.99 | 4.19 |

2 | cyclobutane | 12.51 | 15.29 | -2.78 | 14.64 | -2.13 | 1.77 | 10.74 |

3 | spiropentane | 40.6 | 48.20 | -7.60 | 43.20 | -2.60 | 49.42 | -8.82 |

4 | methylcyclobutane | 36.3 | 38.57 | -2.27 | 38.83 | -2.53 | 33.49 | 2.81 |

5 | cyclopentane | 49.262 | 48.20 | 1.06 | 43.20 | 6.06 | 52.5 | -3.24 |

6 | 1,1-dimethylcyclopropane | 20.63 | 24.92 | -4.29 | 26.19 | -5.56 | 23.95 | -3.32 |

7 | cis-1,2-dimethylcyclopropane | 37.03 | 31.74 | 5.29 | 31.66 | 5.37 | 30.15 | 6.88 |

8 | ethylcyclopropane | 36 | 38.57 | -2.57 | 37.95 | -1.95 | 37.46 | -1.46 |

9 | bicyclo[3.1.0]hexane | 79.2 | 85.14 | -5.94 | 73.57 | 5.63 | 85.82 | -6.62 |

10 | 1,1-dimethylcyclobutane | 56 | 55.03 | 0.97 | 53.43 | 2.57 | 54.31 | 1.69 |

11 | cis-1,2-dimethylcyclobutane | 68 | 61.85 | 6.15 | 62.67 | 5.33 | 62.41 | 5.59 |

12 | tras-1,2-dimethylcyclobutane | 60 | 61.85 | -1.85 | 62.67 | -2.67 | 62.41 | -2.41 |

13 | cis-1,3-dimethylcyclobutane | 60.5 | 61.85 | -1.35 | 61.01 | -0.51 | 59.56 | 0.94 |

14 | tras-1,3-dimethylcyclobutane | 57.5 | 61.85 | -4.35 | 61.01 | -3.51 | 59.56 | -2.06 |

15 | cyclohexane | 80.738 | 75.50 | 5.24 | 76.05 | 4.68 | 84.36 | -3.62 |

16 | methylcyclopentane | 71.812 | 68.68 | 3.14 | 69.84 | 1.98 | 75.98 | -4.17 |

17 | 1,1,2-trimethylcyclopropane | 52.48 | 48.20 | 4.28 | 50.35 | 2.13 | 54.66 | -2.18 |

18 | cis,cis-1,2,3,-trimethylcyclopropane | 71 | 55.03 | 15.97 | 71.01 | -0.01 | 61.37 | 9.63 |

19 | cis,trans-1,2,3,-trimethylcyclopropane | 66 | 55.03 | 10.97 | 55.10 | 10.90 | 61.37 | 4.63 |

20 | cis-1-ethyl-2-ethylcyclopropane | 70 | 91.96 | -21.96 | 70.495 | -0.495 | 64.86 | 5.14 |

21 | propylcyclopropane | 68.5 | 68.68 | -0.18 | 68.11 | 0.39 | 72.82 | -4.32 |

22 | isopropylcyclopropane | 58.34 | 61.85 | -3.51 | 62.15 | -3.81 | 63.18 | -4.84 |

23 | bicyclo[3.2.0]heptane | 109.3 | 115.24 | -5.94 | 103.51 | 5.79 | 112.2 | -2.9 |

24 | bicyclo[4.1.0]heptane | 111.5 | 115.24 | -3.74 | 103.60 | 7.90 | 111.69 | -0.19 |

25 | 2-cyclopropylbutane | 90.98 | 91.96 | -0.98 | 91.89 | -0.91 | 94.75 | -3.77 |

26 | propylcyclobutane | 100.6 | 115.24 | -14.64 | 103.52 | -2.92 | 100.42 | 0.18 |

27 | isopropylcyclobutane | 92.7 | 91.96 | 0.74 | 93.02 | -0.32 | 91.13 | 1.57 |

28 | methylcyclohexane | 100.93 | 98.78 | 2.15 | 100.42 | 0.52 | 104.36 | -3.43 |

29 | 1,1-dimethylcyclopentane | 87.846 | 85.14 | 2.71 | 86.44 | 1.40 | 90.62 | -2.77 |

30 | trans-1,2-dimethylcyclopentane | 91.869 | 91.96 | -0.09 | 93.48 | -1.61 | 98.15 | -6.28 |

31 | cis-1,3-dimethylcyclopentane | 91.725 | 91.96 | -0.24 | 93.68 | -1.95 | 95.52 | -3.79 |

32 | trans-1,3-dimethylcyclopentane | 90.773 | 91.96 | -1.19 | 93.68 | -2.90 | 95.52 | -4.75 |

33 | 1,1,2,2-tetramethylcyclopropane | 75.6 | 64.66 | 10.94 | 75.64 | -0.04 | 74.28 | 1.32 |

34 | 1,1,2,3-tetramethylcyclopropane | 78.5 | 71.49 | 7.01 | 78.08 | 0.42 | 84.01 | -5.51 |

35 | 1-methyl-1-isopropylcyclopropane | 82.1 | 78.31 | 3.79 | 80.28 | 1.82 | 84.83 | -2.73 |

36 | 1,1-dimethylcyclopropane | 88.67 | 85.14 | 3.53 | 84.92 | 3.75 | 92.95 | -4.28 |

37 | 2-methylbicyclo[2.2.1]heptane | 125.8 | 138.53 | -12.73 | 127.90 | -2.10 | 130.33 | -4.53 |

38 | 3,3-dimethylbicyclo[3.1.0]hexane | 115.3 | 124.88 | -9.58 | 119.06 | -3.76 | 110.49 | 4.81 |

39 | 1,1,3,3-tetramethylcyclobutane | 78.2 | 94.77 | -16.57 | 75.30 | 2.90 | 86.57 | -8.37 |

40 | trans-1,2-diethylcyclobutane | 115.5 | 122.07 | -6.57 | 121.40 | -5.90 | 122.24 | -6.74 |

41 | methylcycloheptane | 134 | 128.89 | 5.11 | 131.20 | 2.80 | 133.38 | 0.62 |

42 | 1,1-dimethylcyclohexane | 119.54 | 115.24 | 4.30 | 116.01 | 3.53 | 116.49 | 3.05 |

43 | trans-1,2-imethylcyclohexane | 123.42 | 122.07 | 1.35 | 124.23 | -0.81 | 123.9 | -0.48 |

44 | cis-1,3-dimethylcyclohexane | 120.09 | 122.07 | -1.98 | 123.67 | -3.59 | 121.28 | -1.19 |

45 | trans-1,3-dimethylcyclohexane | 124.45 | 122.07 | 2.38 | 123.67 | 0.78 | 121.28 | 3.17 |

46 | cis-1,4-dimethylcyclohexane | 124.32 | 122.07 | 2.25 | 124.90 | -0.58 | 121.51 | 2.81 |

47 | ethylcyclohexane | 131.78 | 128.89 | 2.89 | 130.24 | 1.54 | 133.19 | -1.41 |

48 | cyclooctane | 151.14 | 135.72 | 15.42 | 137.47 | 13.67 | 145.2 | 5.89 |

49 | 1,1,2-trimethylcyclopentane | 113.73 | 108.42 | 5.31 | 110.08 | 3.65 | 112.39 | 1.34 |

50 | cis,cis-1,1,3-trimethylcyclopentane | 123 | 115.24 | 7.76 | 116.71 | 6.29 | 117 | 6 |

51 | cis,trans-1,1,3-trimethylcyclopentane | 117.5 | 115.24 | 2.26 | 116.71 | 0.79 | 117 | 0.5 |

52 | trans,cis-1,1,3-trimethylcyclopentane | 110.2 | 115.24 | -5.04 | 116.71 | -6.51 | 117 | -6.8 |

53 | 1-ethyl-1-methylcyclopentane | 121.52 | 115.24 | 6.28 | 115.75 | 5.77 | 121.05 | 0.47 |

54 | isopropylcyclopentane | 126.42 | 122.07 | 4.35 | 123.75 | 2.67 | 127.4 | -0.98 |

55 | 1,1,2-trimethyl-2-ethylcyclopropane | 104 | 94.77 | 9.23 | 108.34 | -4.34 | 103.22 | 0.78 |

56 | 1-methyl-1,2-diethylcyclopropane | 108.5 | 108.42 | 0.08 | 110.79 | -2.29 | 114.83 | -6.83 |

57 | 7,7-bicycloylbicyclo[2.2.1]heptane | 143.5 | 124.88 | 18.62 | 141.78 | 1.72 | 143.2 | 0.3 |

58 | 2-ethylbicyclo[2.2.1]heptane | 146.5 | 168.64 | -22.14 | 157.75 | -11.25 | 154.66 | -8.16 |

59 | 4-methylspiro[5.2]octane | 149 | 161.81 | -12.81 | 155.20 | -6.20 | 151.49 | -2.49 |

60 | 1,2-dimethylcycloheptane | 153 | 152.18 | 0.82 | 154.91 | -1.91 | 150.71 | 2.29 |

61 | 1,1,2-trimethylcyclohexane | 145.2 | 138.53 | 6.67 | 140.19 | 5.01 | 136.28 | 8.92 |

62 | 1,1,3-trimethylcyclohexane | 136.63 | 138.53 | -1.90 | 137.22 | -0.59 | 130.74 | 5.88 |

63 | 1,1,4-trimethylcyclohexane | 135 | 138.53 | -3.53 | 141.47 | -6.47 | 131.32 | 3.68 |

64 | 1-ethyl-1-methylcyclohexane | 152.16 | 145.35 | 6.81 | 145.61 | 6.55 | 144.59 | 7.57 |

65 | propylcyclohexane | 156.72 | 159.00 | -2.28 | 160.30 | -3.58 | 159.77 | -3.06 |

66 | isopropylcyclohexane | 154.76 | 152.18 | 2.59 | 154.45 | 0.31 | 150.6 | 4.16 |

67 | cyclononane | 178.4 | 165.82 | 12.58 | 168.18 | 10.22 | 171.95 | 6.45 |

68 | 1,1,2,2-tetramethylcyclopentane | 135 | 124.88 | 10.12 | 129.67 | 5.33 | 124.67 | 10.36 |

69 | 1,1,3,3--tetramethylcyclopentane | 117.96 | 124.88 | -6.92 | 125.09 | -7.13 | 115.29 | 2.67 |

70 | cis-1,2-dimethyl-1-ethylcyclopentane | 143 | 138.53 | 4.47 | 139.53 | 3.47 | 140.15 | 3.15 |

71 | trans-1,2-dimethyl-1-ethylcyclopentane | 142 | 138.53 | 3.47 | 139.53 | 2.47 | 140.15 | 2.15 |

72 | 1-methyl-1-propylcyclopentane | 146 | 145.35 | 0.65 | 145.04 | 0.96 | 147.4 | -1.4 |

73 | 1,1-diethylcyclopentane | 151 | 145.35 | 5.65 | 145.05 | 5.95 | 148.92 | 2.08 |

74 | trans-1,3-dietjhylcyclopentane | 150 | 152.18 | -2.18 | 152.91 | -2.91 | 150.87 | -0.87 |

75 | cis-1-methyl-3-isopropylcyclopentane | 142 | 145.35 | -3.35 | 147.58 | -5.58 | 141.76 | 1.76 |

76 | trans-1-methyl-3-isopropylcyclopentane | 143 | 145.35 | -2.35 | 147.58 | -4.58 | 141.76 | 2.76 |

77 | isobutylcyclopentane | 147.95 | 152.18 | -4.23 | 154.29 | -6.34 | 151.47 | -3.52 |

78 | sec-butylcyclopentane | 154.35 | 152.18 | 2.17 | 153.42 | 0.93 | 153.79 | 0.56 |

79 | 2-cyclopropylhexane | 142.95 | 152.18 | -9.23 | 152.67 | -9.72 | 150.35 | -7.4 |

80 | 3-cyclobutylpentane | 151.5 | 152.18 | -0.68 | 152.06 | -0.56 | 146.12 | 5.38 |

^{2}of 0.961 and 0.977, respectively. Using the LGO cross-validation method, the Eqs 28 and 29 had a overall MAE of 6.429

^{o}C (7.452, 5.766, 7.070, 7.321 and 4.536

^{o}C) and 4.801

^{o}C (5.472, 5.159, 3.539, 5.426 and 4.41

^{o}C), respectively.

no | Cycloalkane | Obsd (^{o}C) | Cald [Eq. 28] | Res. | Cald [Eq. 29] | Res. | Cald [Eq. 1 /25 ] | Res. |
---|---|---|---|---|---|---|---|---|

1 | methylcyclopropane | 0.73 | 8.46 | -7.73 | 8.35 | -7.62 | -2.34 | 3.07 |

2 | trans-1,2-dimethylcyclopropane | 28.21 | 31.74 | -3.53 | 31.66 | -3.45 | 30.15 | -1.94 |

3 | bicyclo[2.2.0]hexane | 80.2 | 85.14 | -4.94 | 73.41 | 6.79 | 78.97 | 1.23 |

4 | ethylcyclobutane | 70.6 | 68.68 | 1.92 | 68.71 | 1.89 | 68.66 | 1.94 |

5 | 1-ethyl-1-methylcyclopropane | 56.77 | 55.03 | 1.74 | 55.46 | 1.31 | 60.36 | -3.59 |

6 | trans-1,2-diethylcyclopropane | 65 | 91.96 | -26.96 | 64.80 | 0.2 | 64.86 | 0.14 |

7 | cycloheptane | 118.79 | 105.61 | 13.18 | 106.76 | 12.03 | 116.11 | 2.68 |

8 | cis-1,2-dymethylcyclopentane | 99.532 | 91.96 | 7.57 | 93.48 | 6.05 | 98.15 | 1.382 |

9 | ethylcyclopentane | 103.46 | 98.78 | 4.68 | 99.56 | 3.90 | 107.67 | -4.204 |

10 | spiro[5.2]octane | 125.5 | 138.53 | -13.03 | 128.38 | -2.88 | 135.02 | -9.52 |

11 | cis-1,2-dimethylcyclohexane | 129.72 | 122.07 | 7.65 | 124.23 | 5.49 | 123.9 | 5.828 |

12 | trans-1,4-dimethylcyclohexane | 119.35 | 122.07 | -2.72 | 124.90 | -5.55 | 121.51 | -2.159 |

13 | 1,1,2-trimethylcyclopentane | 104.89 | 108.42 | -3.53 | 110.08 | -5.18 | 106.86 | -1.967 |

14 | propylcyclopentane | 130.95 | 128.89 | 2.06 | 129.68 | 1.27 | 136.57 | -5.621 |

15 | 2-cyclopropylpentane | 117.74 | 122.07 | -4.33 | 122.09 | -4.35 | 123.66 | -5.92 |

16 | cis-bicyclo[4.3.0]nonane | 166 | 175.46 | -9.46 | 164.38 | 1.62 | 164.59 | 1.41 |

17 | 1,1-dimethyl-2-ethylcyclopentane | 138 | 138.53 | -0.53 | 138.78 | -0.78 | 138.33 | -0.33 |

18 | 1,1-dimethylcyclopentane | 133 | 138.53 | -5.53 | 139.46 | -6.46 | 133.37 | -0.37 |

19 | cis-1,3-diethylcyclopentane | 150 | 152.18 | -2.18 | 152.91 | -2.91 | 150.87 | -0.87 |

20 | butylcyclopentane | 156.6 | 159.00 | -2.40 | 160.22 | -3.62 | 163.27 | -6.67 |

21 | tert-butylcyclopentane | 144.85 | 138.53 | 6.32 | 140.05 | 4.80 | 138.18 | 6.67 |

22 | dicyclobutylmethane | 161.8 | 175.46 | -13.66 | 164.47 | -2.67 | 152.11 | 9.69 |

23 | 1,5-dimethylspiro[3.3]heptane | 132.2 | 154.99 | -22.79 | 135.25 | -3.05 | 142.44 | -10.24 |

24 | 4-methylspiro[5.2]octane | 149 | 161.81 | -12.81 | 155.20 | -6.20 | 151.49 | -2.49 |

25 | 2,6-dimethylbicyclo[3.2.1]octane | 164.5 | 191.92 | -27.42 | 165.4 | -0.90 | 165.41 | -0.91 |

26 | 3,7-dimethylbicyclo[3.3.0]octane | 166 | 191.92 | -25.92 | 166.03 | -0.03 | 165.6 | 0.4 |

^{o}C, respectively. As it can be observe, in both series, the predictability and robustness of the theoretical model was demonstrated.

**Bp (**= -21.10996(±5.894) +0.352115(±0.084)

^{o}C)^{.}

**q**(x) +0.2756648(±0.012)

_{0}^{H}^{.}

**q**(x) +5.420964(±0.218)

_{2}^{.H}

**q**(x) +1.644634(±0.347)

_{1L}^{.E}

**q**(x) +0.041902(±0.012)

_{1L}^{.E}

**q**(x) -0.025834(±0.004)

_{4L}^{H}^{.E}

**q**(x)

_{5L}^{2}=0.9763 F(6.63)=539.43 s=7.6115 MAE=7.34 p<0.0001

^{o}C. The squared correlation coefficient (R

^{2}) for Eq. 30 was 0.981, so this model explained more than 98% of the variance for the experimental Bp values.

^{o}C. The overall MAE was 7.342. Like a more exhaustive corroboration of the predictive power of the model, an external prediction set of 20 aromatic organic compounds was used. The Bp of the compounds included in the external test set was predicted with the same accuracy as compounds in the data set. The statistical parameters for this series were: R= 0.9930, F(1.18)=1274.4 and s=7.8280

^{o}C. These results evidence the good predictive power of the model found. Experimental and calculated Bp of the 20 aromatic compounds is given in Table 15. Considering the full set (training and test set) the correlation coefficients were 0.9884, F(1.88)=3717.5 and s=8.43

^{o}C.

**Table 14.**Experimental and Calculated Values of the Bp of Molecules Included in the Training Set, that Contain Aromatic Cycles in Their Molecular Structure, as Well as Residual of Regression and Cross-Validation.

Compound | Obs. (^{o}C) | Calc. | Res. | R-_{CV} | Compound | Obs. (^{o}C) | Calc. | Res. | R-_{CV} |
---|---|---|---|---|---|---|---|---|---|

Chlorobenzene | 132.00 | 130.79 | 1.21 | 1.34 | Mesitylene | 165.00 | 169.99 | -4.99 | -5.24 |

m-Nitrochlorobenzene | 236.00 | 235.11 | 0.89 | 1.25 | Prehnitene | 205.00 | 191.08 | 13.92 | 15.14 |

p-Nitrochlorobenzene | 239.00 | 237.21 | 1.79 | 2.48 | Isodurene | 197.00 | 191.08 | 5.92 | 6.44 |

Aniline | 184.00 | 187.35 | -3.35 | -3.57 | Durene | 195.00 | 191.08 | 3.92 | 4.26 |

Phenol | 181.00 | 174.56 | 6.44 | 6.78 | Pentamethylbenzene | 231.00 | 212.18 | 18.82 | 21.97 |

o-Cresol | 191.00 | 193.84 | -2.84 | -2.95 | Ethylbenzene | 136.00 | 141.26 | -5.26 | -5.54 |

m-Cresol | 201.00 | 194.85 | 6.15 | 6.34 | n-Propylbenzene | 152.00 | 158.55 | -6.55 | -7.01 |

p-Cresol | 201.00 | 195.22 | 5.78 | 5.95 | tert-Butylbenzene | 169.00 | 179.64 | -10.64 | -11.92 |

o-Toluic Acid | 259.00 | 265.28 | -6.28 | -6.68 | p-Cymene | 177.00 | 179.64 | -2.64 | -2.96 |

m- Toluic Acid | 263.00 | 266.40 | -3.40 | -3.63 | Biphenyl | 255.00 | 257.78 | -2.78 | -3.20 |

p- Toluic Acid | 275.00 | 267.05 | 7.95 | 8.52 | Diphenylmethane | 263.00 | 271.25 | -8.25 | -9.32 |

o-Tolualdehyde | 196.00 | 197.50 | -1.50 | -1.56 | Styrene | 145.00 | 153.11 | -8.11 | -8.65 |

m-Tolualdehyde | 199.00 | 198.25 | 0.75 | 0.78 | Phenylacetaldehyde | 193.00 | 200.62 | -7.62 | -8.65 |

p-Tolualdehyde | 205.00 | 198.68 | 6.32 | 6.61 | Diphenylether | 259.00 | 281.11 | -22.11 | -24.23 |

o-Bromophenol | 194.00 | 191.36 | 2.64 | 2.82 | Benzyl Alcohol | 205.00 | 194.72 | 10.28 | 10.72 |

p-Fluorophenol | 185.00 | 189.05 | -4.05 | -6.64 | α-Phenylethyl Alcohol | 205.00 | 212.19 | -7.19 | -7.54 |

o-Phenylenediamine | 252.00 | 265.08 | -13.08 | -15.96 | β-Phenylethyl Alcohol | 221.00 | 211.43 | 9.57 | 10.37 |

p-Phenylenediamine | 267.00 | 267.44 | -0.44 | -0.53 | α-Picoline | 128.00 | 136.75 | -8.75 | -9.50 |

o-Toluidine | 200.00 | 207.11 | -7.11 | -7.48 | β-Picoline | 143.00 | 139.17 | 3.83 | 4.10 |

m-Toluidine | 203.00 | 207.85 | -4.85 | -5.08 | γ-Picoline | 144.00 | 139.75 | 4.25 | 4.53 |

p-Toluidine | 200.00 | 208.13 | -8.13 | -8.51 | Phthalyc Anhydride | 284.00 | 280.66 | 3.34 | 4.85 |

Benzoic Acid | 250.00 | 245.95 | 4.05 | 4.28 | Naphthalene | 218.00 | 215.18 | 2.82 | 3.23 |

Benzaldehyde | 178.00 | 177.58 | 0.42 | 0.45 | 1-Methylnaphthalene | 241.00 | 236.28 | 4.72 | 5.23 |

m-Anisidine | 251.00 | 244.98 | 6.02 | 6.52 | 2-Methylnaphthalene | 240.00 | 236.28 | 3.72 | 4.12 |

p-Anisidine | 244.00 | 245.78 | -1.78 | -1.93 | 1-Naphtylamine | 301.00 | 292.10 | 8.90 | 9.90 |

o-Nitroaniline | 284.00 | 287.79 | -3.79 | -5.32 | 2-Naphtylamine | 294.00 | 294.61 | -0.61 | -0.69 |

N-Methylaniline | 196.00 | 184.00 | 12.00 | 12.38 | 1-Naphthol | 280.00 | 277.96 | 2.04 | 2.20 |

Acetophenone | 202.00 | 196.57 | 5.43 | 5.65 | 2-Naphthol | 286.00 | 281.38 | 4.62 | 5.04 |

Benzophenone | 308.00 | 310.04 | -2.04 | -2.33 | Phenylthiol | 169.50 | 157.48 | 12.02 | 12.85 |

Benzoyl Chloride | 197.00 | 200.84 | -3.84 | -4.08 | 9,10-Anthraquinone | 380.00 | 374.99 | 5.01 | 9.29 |

o-Xylene | 144.00 | 148.89 | -4.89 | -5.12 | Pyrrole | 130.00 | 120.91 | 9.09 | 10.14 |

m-Xylene | 139.00 | 148.89 | -9.89 | -10.35 | Pyridine | 115.00 | 120.15 | -5.15 | -5.75 |

p-Xylene | 138.00 | 148.89 | -10.89 | -11.40 | Furfuryl Alcohol | 171.00 | 175.81 | -4.81 | -5.42 |

1, 2, 3-Trimethyl benzene | 176.00 | 169.99 | 6.01 | 6.32 | Phenylacetic Acid | 266.00 | 275.84 | -9.84 | -12.57 |

Pseudocumene | 169.00 | 169.99 | -0.99 | -1.04 | Cathechol | 245.00 | 237.21 | 7.79 | 8.70 |

#### Colinearity between variables and redundancy of information

**Table 15.**Experimental and Calculated Values of the Bp of Molecules, Included in the Test Set, that Contain Aromatic Cycles in their Molecular Structure as Well as Residual of Regression.

Compound | Obs. (^{o}C) | Cal. | Res. | Compound | Obs. (^{o}C) | Cal. | Res. |
---|---|---|---|---|---|---|---|

o-Chlorotoluene | 159.00 | 150.17 | 8.83 | sec-butylbenzene | 173.50 | 172.02 | 1.48 |

m-Chlorotoluene | 162.00 | 151.12 | 10.88 | tert-butylbenzene | 284.00 | 284.72 | -0.72 |

p-Chlorotoluene | 162.00 | 151.48 | 10.52 | Cinnamylic Alcohol | 257.50 | 239.34 | 18.16 |

o-Nitrobenzene | 245.00 | 229.54 | 15.46 | 1,4-Dihidronaphthalene | 212.00 | 199.52 | 12.48 |

m-Chlorophenol | 214.00 | 196.98 | 17.02 | Isoquinoline | 243.00 | 222.61 | 20.39 |

m-Phenylendiamine | 287.00 | 266.86 | 20.14 | Phenanthrene | 340.00 | 323.67 | 16.33 |

o-Chloroaniline | 209.00 | 207.48 | 1.52 | Thiophene | 84.00 | 90.31 | -6.31 |

m-Nitroaniline | 307.00 | 292.21 | 14.79 | m-Bromophenol* | 236.00 | 194.79 | 41.21 |

N,N-Dimethylaniline | 194.00 | 182.57 | 11.43 | o-Anisidine* | 225.00 | 241.99 | -16.99 |

Diphenylaniline | 302.00 | 301.00 | 1.00 | p-Nitroaniline* | 232.00 | 293.93 | -61.93 |

n-Propylbenzene | 159.00 | 154.73 | 4.27 | Hexamethylbenzene* | 264.00 | 233.28 | 30.72 |

n-Butylbenzene | 183.00 | 168.20 | 14.80 | Furan* | 32.00 | 105.28 | -73.28 |

Isobutylbenzene | 171.00 | 173.72 | -2.72 |

^{*}Compound detected as an outlier in the training set.

^{2}of 0.9932. To solve this problem Randić proposed a procedure of orthogonalization of molecular descriptors that have been applied with much success to QSPR and QSAR studies [62,63,64,65,66]. The orthogonalization of molecular descriptors is an approach in which molecular descriptors are transformed in such a way that they do not mutually correlate. The nonorthogonal descriptors and the derived orthogonal descriptors both contain the same information, which results in the same statistical parameters of the QSAR models [62,63,64,65,66]. However, the coefficient of the QSAR model based on orthogonal descriptors are stable to the inclusion of novel descriptors, which permits to interpret the regression terms and evaluate the role of individual descriptors to the QSAR model.

**Table 16.**The squared correlation matrix showing covariance (r

^{2}) among the topological descriptors (Total and local quadratic indices) used in the regression analysis for 70 compounds.

^{e}q_{2}(x) | ^{He}q_{1L}(x) | ^{Ee}q_{1L}(x) | ^{Ee}q_{5L}(x) | ^{e}q_{0}^{H}(x) | ^{Ee}q_{4L}^{H}(x) |
---|---|---|---|---|---|

1.0000 | 0.1824 | 0.4142 | -0.3593 | -0.8106 | -0.1738 |

1.0000 | 0.3980 | 0.1503 | -0.0116 | -0.4667 | |

1.0000 | -0.2225 | -0.2098 | -0.6433 | ||

1.0000 | 0.1378 | -0.5776 | |||

1.0000 | 0.1826 | ||||

1.0000 |

Descriptors | Multiple R | Multiple R-square | R-square change | Partial Correlation. | Tolernce | R-square |
---|---|---|---|---|---|---|

^{e}q_{2}(x) | 0.8063 | 0.6501 | 0.6501 | 0.9421 | 0.2060 | 0.7940 |

^{He}q_{1L}(x) | 0.9653 | 0.9317 | 0.2817 | 0.9527 | 0.6936 | 0.3064 |

^{Ee}q_{1L}(x) | 0.9775 | 0.9555 | 0.0238 | 0.5129 | 0.0366 | 0.9634 |

^{Ee}q_{5L}(x) | 0.9865 | 0.9732 | 0.0176 | -0.6647 | 0.0346 | 0.9654 |

^{e}q_{0}^{H}(x) | 0.9885 | 0.9772 | 0.0040 | 0.4687 | 0.2657 | 0.7343 |

^{Ee}q_{4L}^{H}(x) | 0.9904 | 0.9809 | 0.0037 | 0.4046 | 0.0221 | 0.9779 |

#### Interpretation of QSPR models

**Bp**=

**f**(Molecular Weight, H-Bonding Capacity, Dipole Moment,

Molecular Branching)

**(x) and**

^{H}q_{1L}**(x),**

^{E}q_{1L}**(x),**

^{E}q_{5L}**(x) are in relation with the H-bonding capacity (hydrogen atoms as donors and acceptors, respectively). The coefficients of these variables in the Eq. 31 are positive; only local “heteroatoms” quadratic indices of fifth order [**

^{E}q_{4L}^{H}**(x)] have a negative contribution to the property. This is a logical result because when the number of hydrogen atoms bonded to heteroatoms in molecules is increased then the Bp increases also, because the possibility of intermolecular H-bonding increases with the increase of H-X groups (O, N and S) in molecules. In this sense, the “protonic” quadratic indices of first order [**

^{E}q_{5L}**(x)] are the sum of all possible products of electronegativity of the hydrogen atoms and heteroatoms bonded to them. If X is O, N or S atom, then values of this index increase in the same order, because the electronegativity of these atoms decreases from oxygen atom until the sulfur atom. For this reason, this index is an indicative of the number and type of hydrogen atom linked to heteroatoms.**

^{H}q_{1L}**(x),**

^{E}q_{1L}**(x) and**

^{E}q_{5L}**(x) also are in relation with molecular charge, that is to say, these indices are variables that parameterize to the molecular dipole moment. Finally, molecular weight is described for total quadratic indices [**

^{E}q_{4L}^{H}**q**(x) and

_{2}**q**(x)], suppressing and including hydrogen atoms in molecular pseudograph, respectively. For example, the

_{0}^{H}**q**(x) possesses positive contribution to the Bp due to this molecular descriptor is the sum of the squared of all posible products of the electronegativity of all atoms in the molecule, which is an indicative of the molecular size that increase with the number (n) of atoms in the molecule. The other molecular descriptor [

_{0}^{H}**q**(x)] is related with the possible effect of this variable on molecular weight, size and molecular branching. That is, this variable is a good choice to describe the Bp defined by the combination of molecular weight and branching. This influence is demonstrated by the positive contribution of this index to the studied property.

_{2}_{L}stands for the corresponding atom.

**Table 18.**Molecule of 1-methyl-1,2-diethylcyclopropane with the Following Atom Numbering and Their Total and Local (Atom) Quadratic Indices.

Atom (f) | q(x, f)_{0L} | q(x, f)_{1L} | q(x, f)_{2L} | q(x, f)_{14L} | q(x, f)_{15L} | BpA [0C; (Eq. 28)] | Bp_{B} [^{o}C; (Eq. 29) |

a | 6.9169 | 27.6676 | 55.3352 | 3605884 | 9077470 | 47.07 | 34.90 |

b | 6.9169 | 20.7507 | 55.3352 | 3153885 | 7816879 | 25.19 | 20.34 |

c | 6.9169 | 13.8338 | 48.4183 | 2717007 | 6759769 | 6.73 | 8.92 |

e | 6.9169 | 13.8338 | 34.5845 | 1744048 | 4293673 | 13.55 | 13.68 |

f | 6.9169 | 6.9169 | 13.8338 | 687788.9 | 1744048 | 1.91 | 7.30 |

d | 6.9169 | 6.9169 | 27.6676 | 1462530 | 3605884 | -4.91 | 2.01 |

g | 6.9169 | 13.8338 | 27.6676 | 1493988 | 3759467 | 16.96 | 16.56 |

h | 6.9169 | 6.9169 | 13.8338 | 605581.5 | 1493988 | 1.91 | 7.09 |

Total | 55.3352 | 110.6704 | 276.676 | 15470712 | 38551176 | 108.42 | 110.79 |

**A**(Eq. 28) and

**B**(Eq. 29), then the atom contribution for each specific atom is obtained:

**Bp**(a)= (-105.146/8) +3.1629

_{A}^{.}

**q**(x, a)–0.4933

_{1L}^{.}

**q**(x, a)=47.07

_{2L}^{o}C

**Bp**(a)=(-108.197/8) +1.6358

_{B}^{.}

**q**(x, a) +2.038

_{0L}^{.}

**q**(x, a)–0.3016

_{1L}^{.}

**q**(x, a)

_{2L}-1.75x10

^{-5.}

**q**(x, a) +6.42x10

_{14L}^{-6.}

**q**(x, a)=34.90

_{15L}^{o}C

**Bp**(b)= (-105.146/8) +3.1629

_{A}^{.}

**q**(x, b)–0.4933

_{1L}^{.}

**q**(x, b)=25.19

_{2L}^{o}C

**Bp**(b)= (-108.197/8) +1.6358

_{B}^{.}

**q**(x, b)+2.038

_{0L}^{.}

**q**(x, b)–0.3016

_{1L}^{.}

**q**(x, b)

_{2L}-1.75x10

^{-5.}

**q**(x, b) +6.42x10

_{14L}^{-6.}

**q**(x, b)=20.34

_{15L}^{o}C

**Bp**(c)= (-105.146/8) +3.1629

_{A}^{.}

**q**(x, c)–0.4933

_{1L}^{.}

**q**(x, c)=6.73

_{2L}^{o}C

**Bp**(c)= (-108.197/8) +1.6358

_{B}^{.}

**q**(x, c)+2.038

_{0L}^{.}

**q**(x, c)–0.3016

_{1L}^{.}

**q**(x, c)

_{2L}-1.75x10

^{-5.}

**q**(x, c) +6.42x10

_{14L}^{-6.}

**q**(x, c)=8.92

_{15L}^{o}C

**Bp**(d)= (-105.146/8) +3.1629

_{A}^{.}

**q**(x, d)–0.4933

_{1L}^{.}

**q**(x, d)=-4.91

_{2L}^{o}C

**Bp**(d)= (-108.197/8) +1.6358

_{B}^{.}

**q**(x, d)+2.038

_{0L}^{.}

**q**(x, d)–0.3016

_{1L}^{.}

**q**(x, d)

_{2L}-1.75x10

^{-5.}

**q**(x, d) +6.42x10

_{14L}^{-6.}

**q**(x, d)

_{15L}**=**13.68

^{o}C

**Bp**(e)= (-105.146/8) +3.1629

_{A}^{.}

**q**(x, e)–0.4933

_{1L}^{.}

**q**(x, e)=13.55

_{2L}^{o}C

**Bp**(e)= (-108.197/8) +1.6358

_{B}^{.}

**q**(x, e) +2.038

_{0L}^{.}

**q**(x, e)–0.3016

_{1L}^{.}

**q**(x, e)

_{2L}-1.75x10

^{-5.}

**q**(x, e) +6.42x10

_{14L}^{-6.}

**q**(x, e)=13.68

_{15L}^{o}C

**Bp**(f)= (-105.146/8) +3.1629

_{A}^{.}

**q**(x, f)–0.4933

_{1L}^{.}

**q**(x, f)=1.91

_{2L}^{o}C

**Bp**(f)= (-108.197/8) +1.6358

_{B}^{.}

**q**(x, f)+2.038

_{0L}^{.}

**q**(x, f)–0.3016

_{1L}^{.}

**q**(x, f)

_{2L}-1.75x10

^{-5.}

**q**(x, f) +6.42x10

_{14L}^{-6.}

**q**(x, f)=7.30

_{15L}^{o}C

**Bp**(g)= (-105.146/8) +3.1629

_{A}^{.}

**q**(x, g)–0.4933

_{1L}^{.}

**q**(x, g)=16.96

_{2L}^{o}C

**Bp**(g)= (-108.197/8) +1.6358

_{B}^{.}

**q**(x, g)+2.038

_{0L}^{.}

**q**(x, g)–0.3016

_{1L}^{.}

**q**(x, g)

_{2L}-1.75x10

^{-5.}

**q**(x, g) +6.42x10

_{14L}^{-6.}

**q**(x, g)=16.56

_{15L}^{o}C

**Bp**(h)= (-105.146/8) +3.1629

_{A}^{.}

**q**(x, h)–0.4933

_{1L}^{.}

**q**(x, h)=1.91

_{2L}^{o}C

**Bp**(h)= (-108.197/8) +1.6358

_{B}^{.}

**q**(x, h)+2.038

_{0L}^{.}

**q**(x, h)–0.3016

_{1L}^{.}

**q**(x, h)

_{2L}-1.75x10

^{-5.}

**q**(x, h) +6.42x10

_{14L}^{-6.}

**q**(x, h)=7.09

_{15L}^{o}C

**Bp**(Molecule)=-105.146+3.1629

_{A}^{.}

**q**(x)–0.4933

_{1}^{.}

**q**(x)=108.42

_{2}^{ o}C

**Bp**(Molecule)=-108.197+1.6358

_{B}^{.}

**q**(x)+2.038

_{0}^{.}

**q**(x)–0.3016

_{1}^{.}

**q**(x)-1.75x10

_{2}^{-5.}

**q**(x)

_{14}+6.42x10

^{-6.}

**q**(x)=110.79

_{15}^{o}C

**q**(x, f) [

_{0L}**q**(x, e-f)=6.9169+0.9169 and

_{0L}**q**(x, g-h)=6.9169+0.9169] and

_{0L}**q**(x, f) [

_{1}_{L}**q**(x, e-f)=13.8338+6.9169 and

_{1L}**q**(x, g-h)=13.8338+6.9169] had the same value for both ethyl fragments; but the values of the other molecular descriptors included in the obtained models (Eq. 28 and Eq. 29) are not the same; for example:

_{1L}**q**(x, f) [

_{2L}**q**(x, e-f)=34.5845+13.8338 and

_{2L}**q**(x, g-h)=27.6676+13.8338]. In this case, the difference is in relation with the different values of the local qudratic indices of e and g atom, which is logic because the topologic enviroment (in two steps) is not the same for both atoms. Notice that the f and h atoms have the same value for local qudratic indices and their atom contribution in the ethyl fragment is the same [

_{2L}**q**(x, f)=

_{2L}**q**(x, h)=13.8338].The magnitude of the local quadratic indices increases as the order of the index increases as a consequence of the greater amount of structural information contained in higher order local quadratic indices. For intance,

_{2L}**q**(x, e-f) and

_{14; 15L}**q**(x, g-h) contain more information about both ethyl fragment (on the atom that constitute the fragment and on theirs molecular enviroment), than the previous one.

_{14; 15L}## Conclusions

**E (**molecular vector space), whose elements are organic molecules, was defined as a “direct sum” of different ℜ

^{i}spaces.

**M**) of the molecular pseudograph and canonical bases are selected as the quadratic forms’ matrices and bases, respectively. This molecular TIs has been implemented in computer in the

**TOMO-COMD**software, with the aim of creating a new calculation method. Specifically, the electronegativities of the atoms were used as atomic property. These indices were generalized to “higher analogues (higher order)” as number sequences, with the aim of creating a family of descriptors that constitute a tool of great utility for drug design and bioinformatic studies. In addition, this paper introduces a local approach for molecular quadratic indices. The local definition of these indices allows obtaining these descriptors for an atom or a fragment in study, which can be used in the description of molecular properties that are greatly related with the contribution of this portion. This way, for example, these local indices are of great importance in the modeling of properties of molecules that contain heteroatoms in their structure.

## Acknowledgements

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**MDPI and ACS Style**

Ponce, Y.M. Total and Local Quadratic Indices of the Molecular Pseudograph's Atom Adjacency Matrix: Applications to the Prediction of Physical Properties of Organic Compounds. *Molecules* **2003**, *8*, 687-726.
https://doi.org/10.3390/80900687

**AMA Style**

Ponce YM. Total and Local Quadratic Indices of the Molecular Pseudograph's Atom Adjacency Matrix: Applications to the Prediction of Physical Properties of Organic Compounds. *Molecules*. 2003; 8(9):687-726.
https://doi.org/10.3390/80900687

**Chicago/Turabian Style**

Ponce, Yovani Marrero. 2003. "Total and Local Quadratic Indices of the Molecular Pseudograph's Atom Adjacency Matrix: Applications to the Prediction of Physical Properties of Organic Compounds" *Molecules* 8, no. 9: 687-726.
https://doi.org/10.3390/80900687