$Q_8$-MAGIC LABELING OF SOME GRAPHS AND ITS SUBDIVISION GRAPHS

Abstract

Let $Q_8=\{ \pm 1, \pm i, \pm j, \pm k\}$ be the quaternion group with identity element 1. Then a graph $G=(V(G), E(G))$ with $p$ vertices and $q$ edges is said to be $Q_8$-magic if there exist two maps $f: E(G) \rightarrow N_q$ and $g: E(G) \rightarrow Q_8 \backslash\{1\}$ such that the map $f$ is bijective and the map $\ell^*(v): V(G) \rightarrow Q_8$ defined by $\mathrm{I}^*(u)=$ $\prod_{e \in N^*(u)}(f(e), g(e))$ is a constant map, where $N^*(u)$ is the set of all edges incident with $u$. The map $\ell^*$ is called a $Q_8$-magic labeling of $G$. In this paper, we investigate $Q_8$-magic labeling of some graphs and its subdivision graphs. Also, we classify these graphs according to the magic constant.

Received: June 20, 2022
Accepted: September 13, 2022