Path Related Even Sum Graphs

Abstract

Graph theory has become one of the most rapidly growing areas in Mathematics. Graph theory has independently discovered many timers through some puzzles that arose from the physical world consideration of chemical isomers, electrical networks, etc. There are several areas of Graph theory which have received good attention from mathematics. Labeling of graphs is one of the most interesting problem in the area of Graph theory. Graph Labeling was first introduced by Alexandra Rosa during 1960 where the vertices and edges are assigned real; values or subsets of a set subject to certain conditions. In Graph Labeling, vast amount of literature is available. Labeled graphs serve as useful models for a broad range of applications. The concept of Sum Graphs and  Integral Sum Graphs was introduced by F. Harary. A sum graph is a graph for which there is a labeling of its vertices with distinct positive integers so that two vertices are adjacent if and only if the sum of their labels is the label of another vertex. The minimum number of isolated vertices required to make the graph G, a sum graph is called the sum number of G and is denoted by σ(G). Integral sum graphs are defined similarly, except that the labels may be any integers. The minimum number of isolated vertices required to make the graph G, an integral sum graph is called the integral sum number of G and is denoted by ξ(G)