Cycles and Paths Related Vertex-Equitable Graphs

A vertex labeling ξ of a graph χ is referred to as a ‘vertex equitable labeling (VEq.)’ if the induced edge weights, obtained by umming the labels of the end vertices, satisfy the following condition: the absolute difference in the number of vertices v and u with labels ξ(v)=a and ξ(u)=b (where a, b∈Z) is approximately 1, considering a given set A that consists of the first ⌈q2⌉ non egative integers. A graph χ 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.