Line graph characterization of the order supergraph of a finite group

Abstract

The power graph $\mathcal{P}(G)$ is the simple undirected graph with group elements as a vertex set and two elements are adjacent if one of them is a power of the other. The order supergraph $\mathcal{S}(G)$ of the power graph $\mathcal{P}(G)$ is the simple undirected graph with vertex set $G$ in which two vertices $x$ and $y$ are adjacent if $o(x)|o(y)$ or $o(y)|o(x)$. In this paper, we classify all the finite groups $G$ such that the order supergraph $\mathcal{S}(G)$ is the line graph of some graph. Moreover, we characterize finite groups whose order supergraphs are the complement of line graphs.

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