A New Graph over Semi-Direct Products of Groups

Abstract

In this paper, by establishing a new graph $\Gamma (G)$ over the semi-direct product of groups, we will first state and prove some graph-theoretical properties, namely, diameter, maximum and minimum degrees, girth, degree sequence, domination number, chromatic number, clique number of $\Gamma (G)$. In the final section we will show that $\Gamma (G)$ is actually a perfect graph.