On the ψk-polynomial of graph

In this paper, we introduce a new distance-based topological index of a graph G, called a k-distance degree index. It is defined as N k (G) = diam(G) ∑ k=1 (∑ v∈V (G) d k (v))k, where d k (v) = |N k (v)| = |{u ∈ V (G) : d(v, u) = k}| is the k-distance degree of a vertex v in G, d(u, v) is the distance between vertices u and v in G and diam(G) is the diameter of G. Exact formulas of the N k-index for some well-known graphs are presented. Bounds for N k-index and some other interesting results are established. It is shown that, N k-index of any graph G is an even integer number. In addition, an explicit formulae of a cartesian product of graphs are presented and we apply this result to compute the N k-index of some graphs (of chemical and computer science interest) like hypercube Q d , Hamming graphs H(d, n), nanotube R = P n 2C m and nanotori S = C n 2C m , etc.

Electronic Notes in Discrete Mathematics, 2017

In this paper, we obtain M-polynomial of some graph operations and cycle related graphs. As an application, we compute Mpolynomial of some nanostructures viz., [ , ] nanotube, [ , ] nanotorus, line graph of subdivision graph of [ , ] nanotube and [ , ] nanotorus, Vtetracenic nanotube and V-tetracenic nanotorus. Further, we derive some degree based topological indices from the obtained polynomials.

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].

Binary operations on graphs are studied widely in graph theory ever since each of these operations has been introduced. The neighbourhood polynomial plays a vital role in describing the neighbourhood characteristics of the vertices of a graph. In this study neighbourhood polynomial of graphs arising from the operations like conjunction, join and symmetric difference of certain classes of graphs are calculated and tried to characterize the nature of neighbourhood polynomial.

Oriental Journal Of Chemistry, 2019

In this manuscript we have computed third Zagreb index, first Zagreb polynomial, second Zagreb polynomial, third Zagreb polynomial, hyper Zagreb polynomial, forgotten index, forgotten polynomial, symmetric division index and symmetric division polynomial of Graphene. These quantities are based on degrees of the vertices.

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 ( , ) ( , )

Proyecciones (Antofagasta)

2006

Abstract The Padmakar-Ivan (PI) index of a graph G is defined as PI (G)=[n eu (e| G)+ n ev (e| G)], where n eu (e| G) is the number of edges of G lying closer to u than to v, n ev (e| G) is the number of edges of G lying closer to v than to u and summation goes over all edges of G. In this paper, we define the PI polynomial of a graph and investigate some of the elementary properties of this polynomial and compute it for some well-known graphs. Finally, we generalize some of the properties of Wiener polynomial to PI polynomial.

2021

In mathematical chemistry, molecular structure of any chemical substance can be expressed by a numeric number or polynomial or sequence of numbers which represent the whole graph is called topological index. An important branch of graph theory is the chemical graph theory. Because of their worldwide uses, chemical networks have inspired researchers since their development. Determination of the expressions for topological indices of different derived graphs is a new and interesting problem in graph theory. In this article, some graphs which are derived from Honeycomb structure are studied and found their exact results for some neighbourhood degree-based topological indices. Additionally, a comparison is shown graphically among all the derived indices.

2011

The eccentric connectivity index of a graph G, ξ^ C, was proposed by Sharma, Goswami and Madan. It is defined as ξ^ C (G)=∑ u∈ V (G) degG (u) εG (u), where degG (u) denotes the degree of the vertex x in G and εG (u)= Max {d (u, x)| x∈ V (G)}. The eccentric connectivity polynomial is a polynomial version of this topological index. In this paper, exact formulas for the eccentric connectivity polynomial of Cartesian product, symmetric difference, disjunction and join of graphs are presented.