Cycles and Paths Related Vertex-Equitable Graphs

Abstract

A vertex labeling $\xi$ of a graph $\chi$ is referred to as a ‘vertex equitable labeling (VEq.)’ if the induced edge weights, obtained by summing the labels of the end vertices, satisfy the following condition: the absolute difference in the number of vertices $v$ and $u$ with labels $\xi (v)=a$ and $\xi (u)=b$ (where $a,b\in Z$) is approximately $1$, considering a given set $A$ that consists of the first $\lceil \frac{q}{2}\rceil$ non negative integers. A graph $\chi$ that admits a vertex equitable labeling (VEq.) is termed a ‘vertex equitable’ graph. In this manuscript, we have demonstrated that graphs related to cycles and paths are examples of vertex-equitable graphs.