1. Introduction:
Mathematical chemistry provides tools such as polynomials and functions to capture information hidden in the symmetry of molecular graphs and thus predict properties of compounds without using quantum mechanics. A topological index is a numerical parameter of a graph and depicts its topology. Topological indices describe the structure of molecules numerically and are used in the development of qualitative structure-activity relationships (QSARs). The most commonly known invariants of such kinds are degree-based topological indices. These are actually the numerical values that correlate the structure with various physical properties, chemical reactivity, and biological activities [1–5]. It is an established fact that many properties such as heat of formation, boiling point, strain energy, rigidity, and fracture toughness of a molecule are strongly connected to its graphical structure. Hosoya polynomial, Wiener polynomial [6], plays a pivotal role in distance-based topological indices. A long list of distance-based indices can be easily evaluated from the Hosoya polynomial. A similar breakthrough was obtained recently by Deutsch and Klavzar [7], in the context of de- ˇ gree-based indices. Deutsch and Klavzar [7] introduced ˇ Mpolynomial in, 2015, to play a role, parallel to Hosoya polynomial to determine the closed-form of many degree-based topological indices [8–11]. The real power of M-polynomial is its comprehensive nature containing healthy information about degree-based graph invariants. These invariants are calculated on the basis of symmetries present in the 2dmolecular lattices and collectively determine some properties of the material under observation. Benzenoid hydrocarbons play a vital role in our environment and in the food and chemical industries. Benzenoid molecular graphs are systems with deleted hydrogens. It is a connected geometric figure obtained by arranging congruent regular hexagons in a plane so that two hexagons are either disjoint or have a common edge. This figure divides the plane into one infinite (external) region and a number of finite (internal) regions. All internal regions must be regular hexagons. Benzenoid systems are of considerable importance in theoretical chemistry because they are the natural graph representation of benzenoid hydrocarbons. A vertex of a hexagonal system belongs to, at most, three hexagons. A vertex shared by three hexagons is called an internal vertex [12]. In this paper, we study three benzenoid systems, namely, triangular, hourglass, and jagged-rectangle benzenoid systems.
1.1 Basic Definitions:
In the mathematical fields of graph theory and combinatorics, a matching polynomial (sometimes called an acyclic polynomial) is a generating function of the numbers of matchings of various sizes in a graph. It is one of several graph polynomials studied in algebraic graph theory.
In mathematics, graph theory is the study of graphs, which are mathematical structures used to model pairwise relations between objects. A graph in this context is made up of vertices (also called nodes or points) that are connected by edges (also called links or lines). A distinction is made between undirected graphs, where edges link two vertices symmetrically and directed graphs, where edges link two vertices asymmetrically. Graphs are one of the principal objects of study in discrete mathematics.
A generating function is a way of encoding an infinite sequence of numbers (an) by treating them as the coefficients of a formal power series. This series is called the generating function of the sequence.
In the mathematical discipline of graph theory, a matching or independent edge set in an undirected graph is a set of edges without common vertices. Finding a matching in a bipartite graph can be treated as a network flow problem.
In mathematics, a graph polynomial is a graph invariant whose values are polynomials. Invariants of this type are studied in algebraic graph theory. Important graph polynomials include:
· The characteristic polynomial, based on the graph's adjacency matrix.
· The chromatic polynomial, a polynomial whose values at integer arguments give the number of colorings of the graph with that many colors.
· The dichromatic polynomial, a 2-variable generalization of the chromatic polynomial
· The flow polynomial, a polynomial whose values at integer arguments give the number of nowhere-zero flows with integer flow amounts modulo the argument.
· The (inverse of the) Ihara zeta function, defined as a product of binomial terms corresponding to certain closed walks in a graph.
· The Martin polynomial, used by Pierre Martin to study Euler tours
· The matching polynomials, several different polynomials defined as the generating function of the matchings of a graph.
· The reliability polynomial, a polynomial that describes the probability of remaining connected after independent edge failures
· The Tutte polynomial, a polynomial in two variables that can be defined (after a small change of variables) as the generating function of the numbers of connected components of induced subgraphs of the given graph, parameterized by the number of vertices in the subgraph.
Algebraic graph theory is a branch of mathematics in which algebraic methods are applied to problems about graphs. This is in contrast to geometric, combinatoric, or algorithmic approaches. There are three main branches of algebraic graph theory, involving the use of linear algebra, the use of group theory, and the study of graph invariants.
2. Statement of Topic and Aim:
Topological indices help to predict many chemical and biological properties of chemical structures under study. The aim of this report is to study the molecular topology of some benzenoid systems. M-polynomial has wealth of information about the degree-based topological indices.
3. Literature Review:
In 1996 S. Klavzar and I. Gutman introduced the Schultz molecular topological index (MTI) is compared with the Wiener index (W) of a molecular graph. It is shown that ≤ 4W holds for any (connected) graph Γ, with and denoting the smallest and the largest valency, respectively, of the vertices of Γ. For molecular graphs these bounds can be further improved. For instance, for benzenoid systems we have the following: 4W < MTI < 6.93W. This implies that W and MTI are linearly correlated not only in the case of acyclic molecules (which is a previously known result) but also in the case of molecules with arbitrarily many cycles.
In 2011 H. Deng, J. Yang, and F. Xia investigated that a graph invariant I(G) of a connected graph G=(V,E) contributed by the weights of all edges is defined as I(G)= with the summation over all edges, is the weight of edges connecting vertices of degree i and j, is the number of edges of G connecting vertices of degree i and j. It generalizes Randić index, Zagreb index, sum-connectivity index, G index, ABC index etc.
In 1952 E. Deutsch and S. Klavzar proved that Let G be a graph and let (G), i, j ≥ 1, be the number of edges uv of G such that { (G), (G)} = {i, j}. The M-polynomial of G is introduced with M(G; x, y) = (G). It is shown that degree-based topological indices can be routinely computed from the polynomial, thus reducing the problem of their determination in each particular case to the single problem of determining the M-polynomial.
4. Methodology:
M-polynomial is explained as
M(G, a, b)= (1)
There are some important degree-based topological indices defined, and the first Zagreb index was introduced by Gutman and Trinajstic as follows: ´
(G) = (2)
Gutman and Trinajstic proposed the second Zagreb ´ index in 1972, which is stated as
(G) = (3)
The second modified Zagreb index is defined as
(G) = (4)
Generalized Randic Index:
() (5)
References:
[1] G. R¨ucker and C. R¨ucker, “On topological indices, boiling points, and cycloalkanes,” Journal of Chemical Information and Computer Sciences, vol. 39, no. 5, pp. 788–802, 1999.
[2] S. Klavzar and I. Gutman, “A comparison of the Schultz ˇ molecular topological index with the Wiener index,” Journal of Chemical Information and Computer Sciences, vol. 36, no. 5, pp. 1001–1003, 1996
[3] F. M. Br¨uckler, T. Doˇslic´, A. Graovac, and I. Gutman, “On a class of distance-based molecular structure descriptors,” Chemical Physics Letters, vol. 503, no. 4-6, pp. 336–338, 2011.
[4] H. Deng, J. Yang, and F. Xia, “A general modeling of some vertex-degree based topological indices in benzenoid systems and phenylenes,”,” Computers and Mathematics with Applications, vol. 61, no. 10, pp. 3017–3023, 2011.
[5] H. Zhang and F. Zhang, “/e Clar covering polynomial of hexagonal systems I,” Discrete Applied Mathematics, vol. 69, no. 1-2, pp. 147–167, 1996.
[6] I. Gutman, “Some properties of the Wiener polynomial,” Graph %eory Notes New York, vol. 125, pp. 13–18, 1993.
[7] E. Deutsch and S. Klavzˇar, “M-polynomial and degreebased topological indices,” 2014, https://arxiv.org/abs/ 1407.1592.
[8] M. Munir, W. Nazeer, S. Rafique, and S. M. Kang, “Mpolynomial and related topological indices of nanostar dendrimers,” Symmetry, vol. 8, no. 9, p. 97, 2016.
[9] M. Munir, W. Nazeer, S. Rafique, A. R. Nizami, and S. M. Kang, “M-polynomial and degree-based topological indices of titania nanotubes,” Symmetry, vol. 8, no. 11, p. 117, 2016.
[10] Y. Kwun, M. Munir, W. Nazeer, S. Rafique, and S. M. Kang, “M-polynomial and degree-based topological indices of Vphenylenic nanotubes and nanotori,” Scientific Reports, vol. 7, no. 1, p. 8756, 2017.
[11] M. Munir, W. Nazeer, S. Rafique, A. R. Nizami, and S. M. Kang, “Some computational aspects of triangular boron nanotubes,” Symmetry, vol. 9, no. 1, p. 6, 2016.
ss [12] M. Munir, W. Nazeer, S. Rafique, and S. M. Kang, “Mpolynomial and degree-based topological indices of polyhex nanotubes,” Symmetry, vol. 8, no. 12, p. 149, 2016.
amazing data about M polynomial
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