On the power graph of the direct product of two groups

TL;DR

This paper investigates the structure of power graphs of direct product groups, showing they are not isomorphic to common graph products but are isomorphic to a newly introduced generalized product.

Contribution

It introduces a new generalized product of graphs and proves the power graph of a direct product group is isomorphic to this product.

Findings

Power graph of G1×G2 is not isomorphic to direct, Cartesian, or normal products of P(G1) and P(G2)

Power graph of G1×G2 is isomorphic to a new generalized product of P(G1) and P(G2)

Introduces and characterizes the generalized product of graphs

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 $P(G_2)$.