On the vertex degree polynomial of graphs

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].Publisher's Versio

Matching Transversal Edge Domination in Graphs

Let G =(V,E) be a graph. A subset X of E is called an edge dominating set of G if every edge in E - X is adjacent to some edge in X . An edge dominating set which intersects every maximum matching inG is called matching transversal edge dominating set. The minimum cardinality of a matching transversal edge dominating set is called the matching transversal edge domination number of G and is denoted by γmt(G). In this paper, we begin an investigation of this parameter

Further Results on Designs Arising from Some Certain Corona Graphs

In this paper, we determine the partially balanced incomplete block designs and association scheme which are formed by the minimum dominating sets of the graphs C3°K2, and C4°K2. Finally, we determine the number of minimum dominating sets of graph G = Cn°K2 and prove that the set of all minimum dominating sets of G = Cn°K2 forms a partially balanced incomplete block design with two association scheme

On Reciprocals Leap Indices of Graphs

In the field of chemical graph theory, topological indices are calculated based on the molecular graph of a chemical compound. Topological indices are used in the development of Quantitative structure Activity/Propoerty Relations. To study the physico-chemical properties of molecules most commonly used are the Zagreb indices. In this paper, we introduce reciprocals leap indices as a modified version of leap Zagreb indices. The exact values of reciprocals leap indices of some well-known classes of graphs are calculated. Lower and upper bounds on the reciprocals leap indices of graphs are established. The relationship between reciprocals leap indices and leap Zagreb indices are presented

Minimal, vertex minimal and commonality minimal CN-dominating graphs

We define minimal CN-dominating graph $mathbf {MCN}(G)$, commonality minimal CN-dominating graph $mathbf {CMCN}(G)$ and vertex minimal CN-dominating graph $mathbf {M_{v}CN}(G)$, characterizations are given for graph $G$ for which the newly defined graphs are connected. Further serval new results are developed relating to these graphs

Connected Equitable Domination in Graphs

Abstract Let G = (V, E) be a graph. A subset D of V is called an equitable dominating set of a graph G if for every is the degree of u and deg(v) is the degree of v in G. An equitable dominating set D is said to be a connected equitable dominating set if the subgraph D induced by D is connected. The minimum of the cardinalities of the connected equitable dominating sets of G is called the connected equitable domination number and denoted by γ ce (G) In this paper we introduce the connected equitable domination and connected equitable domatic in a graph, bounds for γ ce (G), d ce (G) and its exact values for some standard graphs are found. Mathematics Subject Classification: 05C6

Connected Injective Domination of Graphs

Let G = (V;E) be a connected graph. A subset S of V is calledinjective dominating set (Inj-dominating set) if for every vertex v 2