Cycles and Paths Related Vertex-Equitable Graphs

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.