# Counting Polynomials in Chemistry: Past, Present, and Perspectives

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

## 1. Introduction

_{2}by Scanrtlebury in 1964 [19], before Hosoya showed that W is the half sum of all entries in [Di] and afterwards proposed the Z index in 1971 [20], which he first attributed to Wiener. The Zagreb index (M

_{1}) followed in 1975 [21], which can be related to the Platt and Gordon–Scantlebury by: F=2N

_{2}=M

_{1}-e. Balaban followed with his Centric index in 1979 [22]. Danail Bonchev, Ovanes Mekenyan, and Milan Randiċ proposed a generalization of the graph center concept to cyclic graphs in 1980 [23,24,25]. A way of classifying TIs is the grade of degeneration, and so we mention another of the first generation, proposed by Schultz in 1989 [26,27]. The beginning of the second generation is marked by the molecular connectivity index of Randiċ in 1975 [28], which was characterized by a very low degeneracy. To name another few: the high-order molecular connectivity indices studied by Kier and Hall in 1975 [29,30,31,32]; the information–theoretic indices of Bonchev and Trinajstić in 1977 [33]; the Merrifield and Simmons indices that were studied in 1980 [34,35,36]; in an effort to decrease degeneracy, Mekenyan and Trinajstić proposed the topological information superindex in 1981 [37]; in 1982, Balaban modified the Randiċ formula and gave rise to the average distance-based connectivity index [38]; the information–theoretic indices of Basak and coworkers in 1983 [39]; the orbital information index for graph connections of Bertz in 1988 [40]; or the electrotopological state indices of Kier and Hall in 1990 [41,42]. One important contribution to the idea of using Eigenvalues as TIs was coined by Lovasz and Pelikan in 1973 [43].

_{N}as a neighbourhood Zagreb index of product graphs [52]; and so on. Many authors have computed indexes for different applications [53,54,55], or analytical expressions for such indexes [56,57,58,59,60]. A number of TIs are based on polynomial coefficients and can be derived directly or by using integrals or derivatives [54,61].

^{SM}database (www.cas.org, accessed on the 18

^{th}of August 2023) currently contains almost 200 million entries. However, it would be ideal to know all their properties. During the 1970s, it became obvious it is not feasible to obtain all their properties experimentally. Gutman exemplified, with the aid of maleic anhydride, that a chemical formula is sometimes better represented by a multigraph rather than by a simple graph [69].

_{i,}we require a matrix whose characteristic polynomial (the ChP discussed in this text) is the same as P

_{i}; based on the definition, the roots of this polynomial are identical to the eigenvalues. If one tries to describe the topology of a molecule, one can store information about the adjacencies (the bonds) between atoms as well as the identities (the atoms). Simplifying by disregarding the bond and atom types, the adjacency matrix ([Ad] a matrix of elements b

_{i,j}= 1 if an edge connects vertices i and j and b

_{i,j}= 0 otherwise) and the identity matrix ([Id]) contain only zeros and ones. Any such or derived square matrix ([Tm]) can be used to construct a counting polynomial that carries features of the originating molecule:

_{i,j}] = k), and “x” represents the roots of the polynomial [66,72]. For example, let there be:

## 2. Characteristic Polynomial

_{1}for example, having 5 vertices/atoms and 5 edges/bonds), $[{Id}_{{G}_{1}}{\displaystyle ]}$ and [${Ad}_{{G}_{1}}]$ are as follows:

_{2}(“2” being the number of vertices an isolated edge can have) and R

_{m}(“m” being the number of vertices a ring is composed of). The number of isolated edges = o and the number of rings = p. There is one restriction in the definition of a Sachs graph: $2\xb7\mathrm{o}+\mathrm{p}\xb7\mathrm{m}=k$, where k = the number of vertices. Let S

_{k}be the set of all Sachs subgraphs (s) with “k” vertices of any graph G, and “a” being the total number of vertices of G. The number of components of the Sachs subgraph is c(s) and the number of rings r(s). The definition of ChP is:

- for k = 0, we count the empty Sachs graph with 0 edges and the result is: $1\xb7{(-1)}^{0}\xb7{2}^{0}\xb7{x}^{a-0}={x}^{a}$;
- for k = 1, there can be no isolated edge with one vertex nor such a ring, and the result is 0.

- denoting $q={\sum}_{\mathrm{s}\in {S}_{k}\left({G}_{1}\right)}{(-1)}^{\mathrm{c}\left(\mathrm{s}\right)}{\xb72}^{\mathrm{r}\left(\mathrm{s}\right)}{\xb7x}^{5-k}$,
- for k = 0, $q={x}^{5}$;
- for k = 2, since this is the case where each edge represents a Sachs subgraph, $q={(-1)}^{1}{\xb72}^{0}{\xb7x}^{5-2}+{(-1)}^{1}{\xb72}^{0}{\xb7x}^{5-2}+{(-1)}^{1}{\xb72}^{0}{\xb7x}^{5-2}+{(-1)}^{1}{\xb72}^{0}{\xb7x}^{5-2}+{(-1)}^{1}{\xb72}^{0}{\xb7x}^{5-2}=5\xb7{(-1)}^{1}{\xb72}^{0}{\xb7x}^{5-2}=5\xb7\left(-1\right)\xb71{\xb7x}^{3}=-5{\xb7x}^{3}$;
- for k = 3 and k = 5, it can be seen in Figure 1g–i that these are not Sachs subgraphs;
- for k = 4, from each Sachs subgraph in Figure 1b–e ${q}_{\left(b\right)}={q}_{\left(c\right)}={q}_{\left(d\right)}={q}_{\left(e\right)}={\left(-1\right)}^{2}{\xb72}^{0}{\xb7x}^{5-4}$ and in Figure 1f ${q}_{\left(f\right)}={(-1)}^{1}{\xb72}^{1}{\xb7x}^{5-4}$ (since c(s) counts both edges and rings); as such for k = 4 $q=4\xb71\xb71\xb7x+1\xb7\left(-1\right)\xb72\xb7x=2\xb7x$.

^{th}powers of these roots is equal to the w

^{th}spectral moment of a graph. The coefficients of the ChP can be deduced from the spectral moments, and vice versa [61].

## 3. Permanental Polynomial

- denoting $q={(-1)}^{k}{\xb7x}^{a-k}{\sum}_{s\in {S}_{k}\left({G}_{1}\right)}{2}^{\mathrm{r}\left(\mathrm{s}\right)}$,
- for k = 0, ${\sum}_{\mathrm{k}=0}\mathrm{q}={\mathrm{x}}^{5}$;
- for k = 2, ${\sum}_{\mathrm{k}=2}\mathrm{q}=5{\xb7\mathrm{x}}^{3}$;
- for k = 3 and k = 5, it can be seen in Figure 1g–i that these are not Sachs subgraphs;

_{1}and G

_{2}(graph 1 and graph 2), share the same permanental polynomial, then G

_{1}is a per-cospectral mate of G

_{2}. A graph G

_{1}is said to be characterized by its permanental polynomial if all the per-cospectral mates of G

_{1}have isomorphic graphs to G

_{1}. Complete graphs, stars, regular complete bipartite graphs, and odd cycles are characterized by their permanental polynomials [89].

_{56}[103].

^{2}, which is possibly a property of some class of structures [98].

## 4. Matching Polynomial

- for k = 0, n
_{i}(k) = 1, as there is one possibility of choosing zero edges, namely ∅; - for k = 1, n
_{i}(k) = 5, the number of edges;

_{i}(k) = 0. As such:

^{0}, acyclic Sachs subgraphs [88]:

_{2}O and the crystal structure of TiF

_{2}[114], with similar work being done by Moldal et al. [115,116]. Fujita [117] developed a versatile restricted-subduced-cycle-index method for generating HP polynomials, as well as the Hosoya indices and MP. Ali et al. computed the closed form of MP of zigzag and rhombic benzenoid systems and presented graphs of their relations with the parameters of the structures [118]. Later, they calculated some standard polynomials associated with topological indices and some degree-based topological indices of hyaluronic acid–curcumin conjugates by using general inverse sum indeg index [54]. Kürkçü et al. have developed a matrix–collocation method to solve stiff fractional differential equations with cubic non-linearity based on MP [119]. Yang derived MP for the benzene ring embedded in the P-type surface network in 2D [120]. Mondal et al. obtained the neighborhood MP (NMP) of the paraline graph of some convex polytopes [121]. Later, they derived the NMP of 3-layered and 4-layered probabilistic neural networks and some structure property models [122]. Rauf et al. evaluated MP and NMP with graphical representations of the structure of graphite carbon nitride [123]. A recent survey can further complete this enumeration [124].

## 5. Hosoya Polynomial

- for k = 0, n
_{i}(k) = 1, as there is one possibility of choosing zero edges, namely ∅; - for k = 1, n
_{i}(k) = 5, the number of edges;

_{i}(k) = 0. As such:

_{1}[128].

_{n}, the dihedral group D

_{2m}, and the generalized quaternion group Q

_{4n}. Abbas et al. investigated Hosoya properties of the commuting graph associated with an algebraic structure developed by the symmetries of regular molecular gones [136]. Chen obtained exact formulas for calculating the Hosoya index of dendrimer nanostars [137]. DeFord proved that the permanent–determinant of an n×n matrix constructed from the adjacency matrix of the tree is equivalent to the Z–index. It can be extended to weighted trees and more general chemical structures [97].

_{n}(n > 2) [143].

## 6. Immanantal Polynomials

_{λ}, not to be confused with the roots, which are denoted by “x” in this chapter) is a natural generalization for the permanental and characteristic polynomials.

_{ij}) be a matrix of order “a”, and “χ” one of the irreducible characters of the symmetric permutation group S

_{a}. The immanant of [M], corresponding to the character “χ” of S

_{a}, is defined as:

_{χ}is the determinant. If χ(τ) = 1 for all “τ”, then d

_{χ}is the permanent [61]. The immanantal polynomial of [M] is ${d}_{\mathrm{c}}(x[Id]-[M])$. If [M] is [Ad] (corresponding to a graph G), we call it the immanantal polynomial of graph G.

_{a}. “ξ” is a conjugated class, the set of all “τ” for a given “a” that has the same cycle structure. The matrix $\mathrm{c}=\mathrm{P}\left(\mathrm{a}\right)\times \mathrm{P}\left(\mathrm{a}\right)$ is the matrix of irreducible characters of S

_{a}, where P(a) is the partition function. The elements of “χ” are χ(λ,ξ) indexed according to a partition “λ” and a conjugated class “ξ”. As such:

^{2}} and {2

^{3}}, respectively. In the last variant, it is sufficient to say they are decreasing. For partition {3,2,1} we have prepared an example in Figure 2e, where we have three connected vertices, another two connected vertices, and an independent one. For partitions {6,1

^{0}}, {5,1

^{1}}, {4,1

^{2}}, … {1

^{6}}, we could write a general form $\mathsf{\lambda}=\left\{r,{1}^{a-r}\right\}$. In the case of {6,1

^{0}}, one should not think of 1

^{0}= 1, but of 6 vertices and an independent vertex counted zero times (∅). When r = 1, the immanant of partition λ

_{1}= {1

^{a}} is the determinant of [Ad], and for r = a one obtains {a,1

^{a-a}} = {a,1

^{0}} = {a}, which is the permanent of [Ad].

_{1}(Figure 1a), for example, which has 5 vertices (a = 5). The non-increasing partitions of 5 are {5}, {4,1}, {3,2}, {3,1,1} (denoted {3,1

^{2}}), {2,2,1} (denoted {2

^{2},1}), {2,1,1,1} (denoted {2,1

^{3}}), and {1,1,1,1,1} (denoted {1

^{5}}), totalling 7 partitions. Therefore, G

_{1}will have a family of 7 immanantal polynomials, one for each partition.

^{5}}, π is obtained—both presented in Chapter 2 and 3, respectively.

_{(a)}], we have:

## 7. Laplacian Polynomial

## 8. Zagreb Polynomials

_{1}in Figure 1a, having 5 edges, we take edge “1”; in this case u = 1 and v = 2. For vertex u = 1, it can be seen in Figure 3a that there are two adjacent edges “1” and “2”. For v = 2 in Figure 3b there are also two adjacent edges, “1” and “3”.

- for k = 1, n
_{u}(k) = 2, n_{v}(k) = 2; - for k = 2, n
_{u}(k) = 2, n_{v}(k) = 2; - for k = 3, n
_{u}(k) = 2, n_{v}(k) = 3; - for k = 4, n
_{u}(k) = 2, n_{v}(k) = 3; - and for k = 5, n
_{u}(k) = 3, n_{v}(k) = 1.

_{1}and general 5th M

_{2}Zagreb polynomials for Dyck-56 Networks [154]. Maji and Ghorai determined the Zagreb polynomials, its copolynomials, and their complement for the w

^{th}generalized transformation graphs [155]. First and second Zagreb indexes are found by differentiating their polynomials at x = 1 [54]. Due to the good correlation between Zagreb indices and chemical properties, a considerable effort has been observed in computing Zagreb polynomials since the first ZgP was defined. This effort can be visualized in Table 1:

## 9. Sextet Polynomial

_{4}) in Figure 4a will take its place.

- for k = 0, n
_{r}(k) = 1, as there is one possibility of choosing zero sextets, namely ∅; - for k = 1, n
_{r}(k) = 5, the number of benzene rings;

## 10. Independence Polynomial

- for k = 0, n
_{ia}(k) = 1, as there is one case of whith zero independent vertices, namely ∅; - for k = 1, n
_{ia}(k) = 5, the number of vertices; - and for k = 2, n
_{ia}(k) = 5 sets of two independent vertices (1,4; 1,5; 2,3; 2,5; 3,5).

## 11. King and Domino Polynomials

- for k = 1, n
_{nk}(k) = 3, the number of cells;

## 12. Cluj Polynomial

- for k = 1, n
_{k at}= 7, counting how many times “1” appears in our matrix; - for k = 2, n
_{k at}= 6, counting how many times “2” appears in our matrix; - for k = 3, n
_{k at}= 6, counting how many times “3” appears in our matrix; - for k = 4, n
_{k at}= 1, since “4” appears once,

_{v}vertex-PadmakarIvan—a pairwise summation; Cluj-Sum; and Cluj-Product—pair-wise product or also Szeged) are derived for tori [200].

## 13. Omega Polynomial

_{5}(between vertex “3” and vertex “6”) and b

_{6}(between vertex “4” and vertex “7”) are called codistant (briefly: b

_{5}co b

_{6}) if the distances d between vertices satisfy:

_{(5)}= (b

_{1}, b

_{2}, …, b

_{p}) be (in general) the set of all edges codistant to b

_{5}. If all elements of CO

_{(5)}also satisfy:

_{(5)}is an orthogonal cut (oc) of G

_{7}. We have, in particular, these oc’s: CO

_{(5)}= (b

_{5}, b

_{6}, b

_{7}), CO

_{(1)}= (b

_{1}, b

_{9}), CO

_{(2)}= (b

_{2}, b

_{8}), CO

_{(3)}= (b

_{3}, b

_{11}), CO

_{(4)}= (b

_{4}, b

_{10}), CO

_{(6)}= (b

_{5}, b

_{6}, b

_{7}), CO

_{(7)}= (b

_{5}, b

_{6}, b

_{7}), CO

_{(8)}= (b

_{2}, b

_{8}), CO

_{(9)}= (b

_{1}, b

_{9}), CO

_{(10)}= (b

_{4}, b

_{10}), and CO

_{(11)}= (b

_{3}, b

_{11}).

_{7}is called a cograph if and only if the edge set B(G

_{7}) is the union of disjoint oc(G

_{7}): CO

_{1}∪ CO

_{2}∪ … ∪ CO

_{p}= B and CO

_{i}∩ CO

_{j}= ∅ for i ≠ j, i,j = 1, 2, ..., p. CO

_{1–5}satisfy these requirements. If any two consecutive edges of such a sequence are codistant (satisfying (25)) and belong to the same face of the covering, the sequence is called a quasi-orthogonal cut “qoc” strip. Our explanation is somewhat shorter and may not cover all possibilities, but the original one can be found in Ref. [203]. A note for non-chemical structures: any other graph taken into consideration needs to be a connected graph.

_{qoc}(k) = the number of qoc’s of length “k”:

_{7}), taking into account disjoint oc’s CO

_{1–5}:

- for k = 0, n
_{qoc}(k) = 0; - for k = 1, n
_{qoc}(k) = 0; - for k = 2, n
_{qoc}(k) = 4, CO_{1–4}; - for k = 3, n
_{qoc}(k) = 1, CO_{5}.

## 14. Wheland Polynomial

^{th}degree of excitation.

_{j}by having a single additional line, and if this line together with its end points divides G

_{i}into G

_{i1}and G

_{i2}, then:

_{11}in Figure 11b):

_{12}is a ring that can be populated with 5 double bonds. Since G

_{13}in Figure 11d is equivalent to G

_{14}in Figure 11e in terms of the Wheland polynomial:

## 15. Discussion

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

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**Figure 1.**(

**a**) Graph theoretical representation of methyl-cyclo-butane (G

_{1}); (

**b**–

**e**) independent edges (also Sachs graphs); (

**f**) independent ring (also a Sachs graph); (

**g**–

**i**) improper Sachs graphs.

**Figure 2.**Representation of partition {4,1} of (G

_{1}): (

**a**) vertex “5” is independent and the rest are connected (bonded); (

**b**) “3” is independent; (

**c**) “2” is independent; (

**d**) “1” is independent; and (

**e**) a representation of partition {3,2,1} from the example with 6 vertices, here benzene.

**Figure 4.**(

**a**) benzo[p,q,r]tetraphene (G

_{4}); (

**b**–

**e**) possibilities of choosing two mutually resonant sextets.

**Figure 5.**(

**a**) anthracene; (

**b**) its polyomino analogue (G

_{5}); (

**c**) one way of arranging a single king; (

**d**) Kekulé pattern corresponding to (

**c**); (

**e**) domino pattern corresponding to (

**c**); (

**f**) the only way one can arrange two non-taking kings; (

**g**) Kekulé pattern corresponding to (

**f**); (

**h**) domino pattern corresponding to (

**f**).

**Figure 6.**(

**a**) Figure 1a (G

_{1}); (

**b**) what remains after deleting the path between vertices “1” and “2”; (

**c**,

**d**) what remains after deleting paths between vertices “1” and “5”.

**Figure 8.**(

**a**) the only structure with no degree of excitation; (

**b**) the only structure with degree of excitation one.

**Figure 9.**(

**a**) the only structure with no degree of excitation; (

**b**–

**d**) three structures with degree of excitation one; (

**e**) the only structure with degree of excitation two.

**Figure 10.**(

**a**,

**b**) the two structures with no degree of excitation; (

**c**–

**e**) three structures with degree of excitation one.

**Figure 11.**(

**a**) similar graph theoretical representation of naphthalene (G

_{7}) to Figure 7; (

**b**) graph theoretical representation of naphthalene without double bonds (G

_{11}); (

**c**) what remains after removing a convenient edge in order to apply Equation (30) (G

_{12}); (

**d**,

**e**) result of dividing G

_{11}in order to apply Equation (30) (G

_{13}and G

_{14}).

Structure Name | Author/s | Reference |
---|---|---|

families of nanotubes | Farahani | [150,156,157] |

Capra-Designed Planar Benzenoid Series | Farahani and Vlad | [158] |

nanostar dendrimers | Husin et al., Siddiqu et al. and Kang et al. | [159,160,161] |

zigzag, rhombic, triangular, hourglass and jagged-rectangle benzenoid systems; Silicon Carbide structures | Kwun et al. | [162,163] |

hetrofunctional dendrimers, triangular benzenoids, and nanocones | Gao et al. | [151] |

the generalized class of carbon nanocones | Noreen and Mahmood | [164] |

some nanostructures | Rehman and Khalid | [165] |

perylenediimide-cored dendrimers | Iqbal et al. | [166] |

Cu_{2}O and TiF_{2} | Yang et al. | [167] |

some dendrimers and polyomino chains | Farooq et al. | [168] |

Sierpiński graphs | Siddiqui | [169] |

remdesivir, chloroquine, hydroxychloroquine and theaflavin | Virk | [170] |

benzene ring implanted in P−type surface structure | Sarkar and Pal | [171] |

hexagonal network HX_{n}, the honeycomb network HC_{n}, the silicate sheet network SL_{n} and the oxide network OX_{n} | Salman et al. | [172] |

benzenoid Triangular system and benzenoid Hourglass system | Chu et al. | [173] |

Silicate Network and Silicate Chain Network | Ghani et al. | [174] |

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Joița, D.-M.; Tomescu, M.A.; Jäntschi, L.
Counting Polynomials in Chemistry: Past, Present, and Perspectives. *Symmetry* **2023**, *15*, 1815.
https://doi.org/10.3390/sym15101815

**AMA Style**

Joița D-M, Tomescu MA, Jäntschi L.
Counting Polynomials in Chemistry: Past, Present, and Perspectives. *Symmetry*. 2023; 15(10):1815.
https://doi.org/10.3390/sym15101815

**Chicago/Turabian Style**

Joița, Dan-Marian, Mihaela Aurelia Tomescu, and Lorentz Jäntschi.
2023. "Counting Polynomials in Chemistry: Past, Present, and Perspectives" *Symmetry* 15, no. 10: 1815.
https://doi.org/10.3390/sym15101815