# On Use of the Variable Zagreb vM2 Index in QSPR: Boiling Points of Benzenoid Hydrocarbons

^{1}

^{2}

^{*}

## Abstract

**:**

^{v}M

_{2}index is introduced and applied to the structure-boiling point modeling of benzenoid hydrocarbons. The linear model obtained (the standard error of estimate for the fit model S

_{fit}=6.8

^{o}C) is much better than the corresponding model based on the original Zagreb M

_{2}index (S

_{fit}=16.4

^{o}C). Surprisingly, the model based on the variable vertex-connectivity index (S

_{fit}=6.8

^{o}C) is comparable to the model based on

^{v}M

_{2}index. A comparative study with models based on the vertex-connectivity index, edge-connectivity index and several distance indices favours models based on the variable Zagreb

^{v}M

_{2}index and variable vertex-connectivity index. However, the multivariate regression with two-, three- and four-descriptors gives improved models, the best being the model with four-descriptors (but

^{v}M

_{2}index is not among them) with S

_{fit}=5

^{o}C, though the four-descriptor model contaning

^{v}M

_{2}index is only slightly inferior (S

_{fit}=5.3

^{o}C).

## Introduction

^{v}M

_{2}index in the structure-boiling point modeling of benzenoid hydrocarbons. We selected benzenoid hydrocarbons because there are several structure-boiling point models of these compounds already published [13,14]. Due to this fact, we were also able to carry out a comparative study of the model based on

^{v}M

_{2}index against the models based on the standard vertex-connectivity index, variable vertex-connectivity index, edge-connectivity index and several distance indices. Since the Zagreb index in its original form was derived using graph-theoretical concepts and terminology [15], we will use these in the present report. Graphs will be generated from molecules in the usual way by replacing atoms with vertices and bonds with edges [16]. Besides, graphs that we will use will represent only carbon skeletons of benzenoid hydrocarbons. Therefore, benzenoid hydrocarbons in this report will be presented as various arrangements of hexagons in the plane.

## The Zagreb M_{2} index and Its Variable Form ^{v}M_{2}

_{2}index together with the Zagreb M

_{1}index appeared in the topological formula for the total π-electron energy of conjugated molecules [17]:

^{v}M

_{2}index is by means of an example. For this purpose we will use a graph G representing the carbon skeleton of naphthalene (see Figure 1).

^{v}M

_{2}index for naphthalene as a function of the variables x and y:

^{v}M

_{2}= 6 (2 + x)

^{2}+ 4 (2 + x) (3 + y) + (3 + y)

^{2}

^{v}M

_{2i}index of a benzenoid hydrocarbon i can be given as:

^{v}M

_{2i}= c

_{1i}(2 + x)

^{2}+ c

_{2i}(2 + x) (3 + y) + c

_{3i}(3 + y)

^{2}

_{i.}Thus, for naphthalene (i=1), bp

_{1}=218°C and c

_{11}, c

_{21}and c

_{31}are respectively 6, 4 and 1. In Table 1 we give bp

_{i}, c

_{1i}, c

_{2i}and c

_{3i}values for 21 benzenoid hydrocarbons whose graphs are given in Figure 2.

**Table 1.**The values of experimental boiling points (bp

_{i}in

^{o}C, i=1,…,21) and coefficients (c

_{1i}, c

_{2i}and c

_{3i}) of the variable Zagreb

^{v}M

_{2}indices of 21 benzenoid hydrocarbons.

Benzenoid hydrocarbon | bp_{i} | ^{v}M_{2} index | ||
---|---|---|---|---|

c_{1i} | c_{2i} | c_{3i} | ||

1 | 218 | 6 | 4 | 1 |

2 | 338 | 7 | 6 | 3 |

3 | 340 | 6 | 8 | 2 |

4 | 431 | 8 | 8 | 5 |

5 | 425 | 7 | 10 | 4 |

6 | 429 | 9 | 6 | 6 |

7 | 440 | 6 | 12 | 3 |

8 | 496 | 7 | 10 | 7 |

9 | 493 | 8 | 8 | 8 |

10 | 497 | 8 | 8 | 8 |

11 | 547 | 6 | 12 | 9 |

12 | 542 | 7 | 10 | 10 |

13 | 535 | 9 | 10 | 7 |

14 | 536 | 8 | 12 | 6 |

15 | 531 | 8 | 12 | 6 |

16 | 519 | 9 | 10 | 7 |

17 | 590 | 6 | 12 | 12 |

18 | 592 | 9 | 10 | 10 |

19 | 596 | 8 | 12 | 9 |

20 | 594 | 8 | 12 | 9 |

_{2i}= 2

^{2}c

_{1i}+ 2·3 c

_{2i}+ 3

^{2}c

_{3i}

_{2}index of a given benzenoid i.It should also be noted that the variable connectivity index

^{v}χ

_{i}is related to the Zagreb

^{v}M

_{2i}index with the same set of coefficients:

^{v}χ

_{i}= c

_{1i}(2 + x)

^{-1 }+ c

_{2i}[(2 + x) (3 + y)]

^{-1/2}+ c

_{3i}(3 + y)

^{-1}

^{v}χ

_{i}= 2

^{-1}c

_{1i}

**+**(2

**·**3)

^{-1/2}c

_{2i}

**+**3

^{-1}c

_{3i}

## Results and Discussion

^{v}M

_{2}index, the values of x were varied in the range between –2 and 2 and values of y were varied in the range between –3 and 3, both in steps of 0.1. This range of x and y values was imposed by the degrees of valences in benzenoid graphs. In non-optimized Zagreb index (M

_{2}), the values of variables x and y are equal to 0.0. We want to see are there optimal values of x and y near their non-optimized values (0.0, 0.0) for which the standard error of estimate of the structure-boiling point model reaches minimum. For each pair (x, y) in the given range, coefficients a

_{0}and a

_{1}in the linear regression model:

_{0}+ a

_{1}

^{v}M

_{2}

_{fit}, the standard error of estimate S

_{fit}and F, the Fisher´s values. S

_{fit}was computed with N and N-I-1 in the denominator, where N is the number of considered benzenoid hydrocarbons and I is the number of descriptors used in the model. In addition, the models were cross(internally)-validated using the leave-one-out method. Statistical parameters for the cross-validated models are denoted by R

_{cv}and S

_{cv}, where cv denotes the cross-validation.

^{v}M

_{2}index with the optimum values of x (0.0) and y (-1.2) is as follows:

^{v}M

_{2}

_{fit}=0.998 S

_{fit}(N)=6.8 S

_{fit}(N-I-1)=7.2 R

_{cv}=0.997 S

_{cv}(N)=8.0 S

_{cv}(N-I-1)=8.4 F=3866

**Figure 3.**Scatter plot between the fit standard error of estimate (S

_{fit}(N)) and x values in the range from –0.4 to +1.1 for the optimum value of y (-1.2)

**Figure 4.**Scatter plot between the fit standard error of estimate (S

_{fit}(N)) and y values in the range from –3 to +1.1 for the optimum value of x (0.0)

**Figure 5.**Scatter plot between the cross validated standard errors of estimate (S

_{cv}(N)) and x values in the range from –0.4 to +1.1 for the optimum value of y (-1.2)

**Figure 6.**Scatter plot between the cross validated standard errors of estimate (S

_{cv}(N)) and y values in the range from –3 to +1.1 for the optimum value of x (0.0)

**Figure 7.**Scatter plots between the experimental (bp

_{exp}) and calculated values of the fit (bp

_{fit}) and cross-validated (bp

_{cv}) models, respectively

**Figure 8.**Scatter plots of fit residuals versus fitted values (bp

_{fit}) and of cross validated residuals versus cross validated values (bp

_{cv}), respectively

_{fit}(N) and x values for the optimum value of y (–1.2) and in Figure 4, we give the scatter plot between S

_{fit}(N) and y values for the optimum value of x (0.0). The results for the fitted models were supported by the results for the cross-validated models (see Figure 5 and Figure 6).

_{2}index:

_{2}

_{fit}=0.986 S

_{fit}(N)=16.4 S

_{fit}(N-I-1)=17.2 R

_{cv}=0.980 S

_{cv}(N)=19.6 S

_{cv}(N-I-1)=20.6 F=656

^{v}χ and the CROMRsel procedure. The following linear model is obtained for the optimum parameters x (0.0) and y (0.5):

^{v}χ

_{fit}=0.998 S

_{fit}(N)=6.8 S

_{fit}(N-I-1)=7.2 R

_{cv}=0.997 S

_{cv}(N)=8.0 S

_{cv}(N-I-1)=8.4 F=3866

_{2}index in building QSPR models, though there are indications [22] that the various variable forms of these two indices lead to the models of the same quality regarding the statistical parameters. However, that is so because both eqs. (5) and (7) are based on the same coefficients c

_{1i}, c

_{2i}and c

_{3i}at constant factors.

_{i}= b

_{0}+ b

_{1}c

_{1i}+ b

_{2}c

_{2i}+ b

_{3}c

_{3i}

_{1i}, c

_{2i}and c

_{3i}are taken from Table 1.

_{1}+ 21.2 (± 0.9) c

_{2}+ 18.4 (± 0.7) c

_{3}

_{fit}=0.998 S

_{fit}(N)=6.8 S

_{fit}(N-I-1)=7.5 R

_{cv}=0.996 S

_{cv}(N)=9.2 S

_{cv}(N-I-1)=10.2 F=1176

_{fit}and S

_{fit}(N) almost identical to those in the models (10) and (12). These parameters would be exactly the same if x and y were obtained more accurately. Because of the greater number of descriptors used in model (14) its S

_{fit}(N-I-1), S

_{cv}(N), S

_{cv}(N-I-1), R

_{cv}and F values are somewhat worse.

_{0}= a

_{0}

_{1}= a

_{1}(2 + x)

^{2}

_{2}= a

_{1}(2 + x) (3 + y)

_{3}= a

_{1}(3 + y)

^{2}

_{fit}=0.997 S

_{fit}(N-I-1)=8.1 F=3045

_{fit}=0.997 S

_{fit}(N-I-1)=7.9 F=3179

^{2}

_{fit}=0.997 S

_{fit}(N-I-1)=8.2 F=1460

^{2}

_{fit}=0.998 S

_{fit}(N-I-1)=7.3 F=1848

_{fit}= 0.998 S

_{fit}(N-I-1)= 7.2 F = 1944

**Δ**/

**D**.

**Δ**is the detour matrix [32,33] and

**D**is the distance matrix [34].

^{v}M

_{2}, χ, ε and three distance indices (the Wiener sum index WS, ws, the detour index ω). The WS index is a Wiener-like index, obtained from the quotient matrix

**D/**

**Δ**[35] and ω is equal to the half-sum of the elements of the detour matrix [32,36]. We considered χ, ε, WS, ws and ω indices because they have been used in the previous structure-boiling point studies of benzenoid hydrocarbons [13,14]. Below we give the best obtained models followed by the best models containing the

^{v}M

_{2}index − in the case of the two-descriptor model the best model contains the

^{v}M

_{2}index:

- (i)
- The two-descriptor modelbp = - 40 (± 22) + 6.42 (± 0.46)
^{v}M_{2}– 0.021 (± 0.014) wsN=21 R_{fit}=0.998 S_{fit}(N)=6.5 S_{fit}(N-I-1)=7.0 R_{cv}=0.996 S_{cv}(N)=8.4 S_{cv}(N-I-1)=8.9 F=2048 - (ii)
- The three-descriptor modelsbp = - 46 (± 18) + 52.7 (± 3.3) ε – 0.167 (± 0.044) ws + 0.037 (± 0.012) ωN=21 R
_{fit}=0.999 S_{fit}(N)=5.3 S_{fit}(N-I-1)=5.9 R_{cv}=0.997 S_{cv}(N)=7.5 S_{cv}(N-I-1)=8.2 F=1936bp = - 47 (± 24) + 37.9 (± 3.4)^{v}M_{2}– 0.32 (± 0.45) ws - 0.024 (± 0.015) WSN=21 R_{fit}=0.998 S_{fit}(N)=6.3 S_{fit}(N-I-1)=7.0 R_{cv}=0.996 S_{cv}(N)=8.6 S_{cv}(N-I-1)=9.4 F=1338 - (iii)
- The four-descriptor modelsbp = - 62 (± 22) + 59.2 (± 5.8) ε – 0.72 (± 0.54) WS – 0.242 (± 0.071) ws + 0.057 (± 0.019) ωN=21 R
_{fit}=0.999 S_{fit}(N)=5.0 S_{fit}(N-I-1)=5.7 R_{cv}=0.997 S_{cv}(N)=7.6 S_{cv}(N-I-1)=8.5 F=1518bp = - 88 (± 26) + 8.62 (± 0.91)^{v}M_{2}– 1.77 (± 0.68) WS – 0.207 (± 0.073) ws + 0.051 (± 0.020) ωN=21 R_{fit}=0.999 S_{fit}(N)=5.3 S_{fit}(N-I-1)=6.1 R_{cv}=0.997 S_{cv}(N)=7.7 S_{cv}(N-I-1)=8.7 F=1330

_{2}index (10) and the variable connectivity index (18) are also very good models with the standard errors of estimate for the fit and cross-validated models of 6.8 (7.2)

^{o}C and 8.0 (8.4)

^{o}C, respectively.

^{v}M

_{2}, ε, WS, ws and ω computed for 21 benzenoid is given in Table 2.

^{v}M_{2} | 1.000 | 0.999 | 0.946 | 0.981 | 0.973 |

ε | 1.000 | 0.936 | 0.983 | 0.974 | |

WS | 1.000 | 0.912 | 0.920 | ||

ws | 1.000 | 0.997 |

^{v}M

_{2},ε;

^{v}M

_{2},ws;

^{v}M

_{2},ω;ε, ws; ε,ω; ws, ω ) or appreciably correlated (

^{v}M

_{2},WS; ε,WS; WS, ws; WS,ω). However, as Randić [e.g., 38] pointed out, the intercorrelation criterion should not be always used for filtering descriptors to be used in building up the QSPR models.

## Conclusions

_{2}index was used in the structure-boiling point modeling of benzenoid hydrocarbons. The obtained model is practically identical to the model based on the variable vertex-connectivity index and this is due to close relationship between the formulas for the two indices.

^{v}M

_{2}index. However, the next best three- and four-descriptor models contain the

^{v}M

_{2}index. The best two-descriptor model contains

^{v}M

_{2}index. The standard errors of estimate for the fit and cross-validated models listed in this report are in the 5.0

^{o}C–9.4

^{o}C range and this is a very good result since it shows that the boiling points of benzenoid hydrocarbons can be predicted within an error range of 0.8– 4.3%.

## Acknowledgments.

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**MDPI and ACS Style**

Nikolića, S.; Miličević, A.; Trinajstić, N.; Jurić, A.
On Use of the Variable Zagreb vM2 Index in QSPR: Boiling Points of Benzenoid Hydrocarbons. *Molecules* **2004**, *9*, 1208-1221.
https://doi.org/10.3390/91201208

**AMA Style**

Nikolića S, Miličević A, Trinajstić N, Jurić A.
On Use of the Variable Zagreb vM2 Index in QSPR: Boiling Points of Benzenoid Hydrocarbons. *Molecules*. 2004; 9(12):1208-1221.
https://doi.org/10.3390/91201208

**Chicago/Turabian Style**

Nikolića, Sonja, Ante Miličević, Nenad Trinajstić, and Albin Jurić.
2004. "On Use of the Variable Zagreb vM2 Index in QSPR: Boiling Points of Benzenoid Hydrocarbons" *Molecules* 9, no. 12: 1208-1221.
https://doi.org/10.3390/91201208