On L h , k -Labeling Index of Inverse Graphs Associated with Finite Cyclic Groups

Abstract

An $L\left(h,k\right)$ -labeling of a graph $G=\left(V,E\right)$ is a function $f:V⟶\left[0,\infty \right)$ such that the positive difference between labels of the neighbouring vertices is at least $h$ and the positive difference between the vertices separated by a distance 2 is at least $k$ . The difference between the highest and lowest assigned values is the index of an $L\left(h,k\right)$ -labeling. The minimum number for which the graph admits an $L\left(h,k\right)$ -labeling is called the required possible index of $L\left(h,k\right)$ -labeling of $G$ , and it is denoted by ${\lambda }_k^h\left(G\right)$ . In this paper, we obtain an upper bound for the index of the $L\left(h,k\right)$ -labeling for an inverse graph associated with a finite cyclic group, and we also establish the fact that the upper bound is sharp. Finally, we investigate a relation between $L\left(h,k\right)$ -labeling with radio labeling of an inverse graph associated with a finite cyclic group.