Some Invariants of Cartesian Product of a Path and a Complete Bipartite Graph

Ramy Shaheen,Qays Alhawat,Suhail Mahfud

doi:10.11648/j.dmath.20190402.11

Abstract

A topological index of graph G is a numerical parameter related to G, which characterizes its topology and is preserved under isomorphism of graphs. Properties of the chemical compounds and topological indices are correlated. In this paper we will compute M-polynomial, first and second Zagreb polynomials and forgotten polynomial for the Cartesian Product of a path and a complete bipartite graph for all values of n and m. From the M-polynomial, we will compute many degree-based topological indices such that general Randic index, inverse Randic index, first and second Zagreb index, modified Zagreb index, Symmetric division index, Inverse sum index augmented Zagreb index and harmonic index for the Cartesian Product of a path and a complete bipartite graph. Also, we will compute the hyper- Zagreb index, the first and second multiple Zagreb index and forgotten index for the Cartesian Product of a path and a complete bipartite graph.

Highlights

Through this paper we consider simple connected graph, i.e. connected without loops and multiple edges
A molecular graph is a simple graph in which atoms and chemical bonds between them are represented by vertices and edges respectively
A topological index is a function that characterizes the topology of the graph

Summary

Introduction

Through this paper we consider simple connected graph, i.e. connected without loops and multiple edges. A graph , with vertex set and edge set is connected if there exists a connection between any pair of vertices in. Most commonly known invariants of such kinds are degree-based topological indices. These are the numerical values that correlate the structure with various physical properties, chemical reactivity and biological activities. Ramy Shaheen et al.: Some Invariants of Cartesian Product of a Path and a Complete Bipartite Graph of degree one and 2 vertices of degree two.

Basic Definitions and Literature Review

Main Results and Discussions

Augmented Zagreb Index