On the power graph of the direct product of two groups

A. K. Bhuniya; Sajal Kumar Mukherjee

paper

On the power graph of the direct product of two groups

arXiv:1606.07258

Abstract

The power graph $P (G)$ of a finite group $G$ is the graph with vertex set $G$ and two distinct vertices are adjacent if either of them is a power of the other. Here we show that the power graph $P (G_{1} \times G_{2})$ of the direct product of two groups $G_{1}$ and $G_{2}$ is not isomorphic to either of the direct, cartesian and normal product of their power graphs $P (G_{1})$ and $P (G_{2})$ . A new product of graphs, namely generalized product, has been introduced and we prove that the power graph $P (G_{1} \times G_{2})$ is isomorphic to a generalized product of $P (G_{1})$ and $P (G_{2})$ .

arXiv comment5 pages