The higher-order matching polynomial of a graph

Scientific Reports, 2019

A topological index of a molecular structure is a numerical quantity that differentiates between a base molecular structure and its branching pattern and helps in understanding the physical, chemical and biological properties of molecular structures. In this article, we quantify four counting polynomials and their related topological indices for the series of a concealed non-Kekulean benzenoid graph. Moreover, we device a new method to calculate the pI and sd indices with the help of theta and omega polynomials. Graph theory has numerous applications in modern chemistry. In chemical graph theory, the vertices and edges respectively represent the atoms and bonds of a molecular structure. To predict the chemical structure using numerical quantity (i.e., topological indices) graph theory plays a vital role. Topological indices have many applications in theoretical chemistry, especially in QSPR/QSAR research. Numerous researchers have conducted studies on topological indices for different graph families; these indices have important chemical significance in the fields of chemical graph theory, molecular topology, and mathematical chemistry. Diudea was the first chemist to consider the subject of computing the topological indices of nanostructures 1-3 . A small particle of an object of intermediate size (between the microscopic and molecular structures of the object) is called a nanoparticle of that object. Nanoparticles are products derived through engineering at the molecular scale. Let G (V, E) be a connected graph with a vertex set V and an edge set E. For any two vertices v 1 and v 2 in G, the distance between v 1 and v 2 is denoted by d (v 1 , v 2 )-the shortest path between v 1 , and v 2 . If e is the edge formed by joining v 1 and v 2 , and f is an edge formed by joining v 3 and v 4 , then e = v 1 v 2 and + 1 and is denoted by 'e co f' . Here, the corelation is symmetric and reflexive but not transitive. Let C (e) = {f ∈ E (G); f co e}: if the 'co' relation is transitive, then the set C (e) is called the orthogonal cut and denoted by co of G. The set of opposite edges that lie along the same face or the same ring, eventually forming a strip of adjacent faces or rings, is called an opposite edge strip and denoted by 'ops' . This concept is also termed a quasi-orthogonal cut, denoted by 'qoc' . Here, the co distance edges are defined within the entire graph G, while 'ops' are defined in the same face or ring. By m (G, c), we mean the number of strips of length c. In this paper, we constructed four polynomials: Omega, Sadhana, Theta and PI. Counting polynomials are those polynomials whose exponent is the extent of a property partition and whose coefficients are the multiplicity of the corresponding partition. We also calculated the topological indices related to these polynomials and formulae. Each counting polynomial represents interesting topological properties of the molecular graph. These polynomials are constructed on the basis of quasi-orthogonal cut edge strips for the series of concealed non-Kekulean benzenoid graphs. The counting polynomials and matching polynomials are useful for topologically describing bipartite structures as well as for counting some single-number descriptors (i.e., the topological indices). The Omega and Theta polynomials count equidistant edges of the graph, while the Sadhana and PI polynomials count nonequidistant edges. Various results related to counting polynomials and topological indices can be found in . The Omega polynomial of a graph G (V, E) is denoted byω (G, x); more information can be found in . The Omega polynomial is defined as ω = ∑ G x m G c x s ( , ) ( , )

A novel graph polynomial, termed as vertex degree polynomial, has been conceptualized, and its discriminating power has been investigated regarding its coefficients and the coefficients of its derivatives and their relations with the physical and chemical properties of molecules. Correlation coefficients ranging from 95% to 98% were obtained using the coefficients of the first and second derivatives of this new polynomial. We also show the relations between this new graph polynomial, and two oldest Zagreb indices, namely the first and second Zagreb indices. We calculate the vertex degree polynomial along with its roots for some important families of graphs like tadpole graph, windmill graph, firefly graph, Sierpinski sieve graph and Kragujevac trees. Finally, we use the vertex degree polynomial to calculate the first and second Zagreb indices for the Dyck-56 network and also for the chemical compound triangular benzenoid G[r].

Symmetry

Topological indices (molecular descriptors) are numerical values of a chemical structure and represented by a graph. Molecular descriptors are used in QSPR/QSAR modeling to determine a chemical structure’s physical, biological, and chemical properties. The cycle graphs are symmetric graphs for any number vertices. In this paper, recently defined neighborhood degree sum-based molecular descriptors and polynomials are studied. NM-polynomials and molecular descriptors of some cycle-related graphs, which consist of the wheel graph, gear graph, helm graph, flower graph, and friendship graph, are computed and compared.

Discrete Mathematics, 2009

We consider the matching polynomials of graphs whose edges have been cyclically labelled with the ordered set of t labels {x 1 ,. . ., x t }. We first work with the cyclically labelled path, with first edge label x i , followed by N full cycles of labels {x 1 ,. . ., x t }, and last edge label x j. Let Φ i,N t+j denote the matching polynomial of this path. It satisfies the (τ, ∆)-recurrence: Φ i,N t+j = τ Φ i,(N −1)t+j −∆ Φ i,(N −2)t+j , where τ is the sum of all non-consecutive cyclic monomials in the variables {x 1 ,. . ., x t } and ∆ = (−1) t x 1 • • • x t. A combinatorial/algebraic proof and a matrix proof of this fact are given. Let G N denote the first fundamental solution to the (τ, ∆)-recurrence. We express G N (i) as a cyclic binomial using the Symmetric Representation of a matrix, (ii) in terms of Chebyshev polynomials of the second kind in the variables τ and ∆, and (iii) as a quotient of two matching polynomials. We extend our results from paths to cycles and rooted trees.

Applied and Computational Mathematics, 2016

Let G be a simple connected graph having vertex set V and edge set E. The vertex-set and edge-set of G denoted by V(G) and E(G), respectively. The length of the smallest path between vertices u,v∈ V(G) is called the distance, d(u,v), between the vertices u,v. Mathematical chemistry is the area of research engaged in new application of mathematics in chemistry. In mathematics chemistry, we have many topological indices for any molecular graph, that they are invariant on the graph automorphism. In this research paper, we computing the Wiener index and the Hosoya polynomial of the Jahangir graphs J 5,m for all integer number m≥3. The Wiener index is the sum of distances between all pairs of vertices of G as W(G)=

2020

There are plenty of topological indices used in chemistry to study the chemical behavior and physical properties of molecular graphs. In the literature, several results are computed for degree based topological indices like “first Zagreb index, second Zagreb index, modified second Zagreb index, generalized Randić index, inverse Randić index, symmetric sum division index, harmonic index, inverse sum index, augmented Zagreb index”. In this paper, we have investigated the aforesaid degree based topological indices for Hanoi graph and generalized wheel graph with the help of M-polynomial.

International Journal of Mathematics and Mathematical Sciences

The study of topological indices in graph theory is one of the more important topics, as the scientific development that occurred in the previous century had an important impact by linking it to many chemical and physical properties such as boiling point and melting point. So, our interest in this paper is to study many of the topological indices “generalized indices’ network” for some graphs that have somewhat strange structure, so it is called the cog-graphs of special graphs “molecular network”, by finding their polynomials based on vertex − edge degree then deriving them with respect to x , y , and x y , respectively, after substitution x = y = 1 of these special graphs are cog-path, cog-cycle, cog-star, cog-wheel, cog-fan, and cog-hand fan graphs; the importance of some types of these graphs is the fact that some vertices have degree four, which corresponds to the stability of some chemical compounds. These topological indices are first and second Zagreb, reduced first and se...

Journal of Chemica Acta

Croatica Chemica Acta

Given a molecular graph G, the Hosoya index Z(G) of G is defined as the total number of the matchings of the graph. Let B n denote the set of bicyclic graphs on n vertices. In this paper, the minimal, the second-, the third-, the fourth-, and the fifth-minimal Hosoya indices of bicyclic graphs in the set B n are characterized.

A topological index is a numerical descriptor of a molecule, based on a certain topological feature of the corresponding molecular graph. In this paper, we explore here some basic mathematical properties and present explicit formulas for the second Hyper-Zagreb coindex under graph operations (disjunction and symmetric difference).