On forbidden subgraphs of main supergraphs of groups

Abstract

In this study, we explore the main supergraph $ \mathcal{S}(G) $ of a finite group $ G $, defined as an undirected, simple graph with a vertex set $ G $ in which two distinct vertices, $ a $ and $ b $, are adjacent in $ \mathcal{S}(G) $ if the order of one is a divisor of the order of the other. This is denoted as either $ o(a)\mid o(b) $ or $ o(b)\mid o(a) $, where $ o(\cdot) $ is the order of an element. We classify finite groups for which the main supergraph is either a split graph or a threshold graph. Additionally, we characterize finite groups whose main supergraph is a cograph. Our classification extends to finite groups $ G $ with $ \mathcal{S}(G) $, a cograph that includes when $ G $ is a direct product of two non-trivial groups, as well as when $ G $ is either a dihedral group, a generalized quaternion group, a symmetric group, an alternating group, or a sporadic simple group.

Keywords

• main supergraph
• finite group
• split graph
• threshold graph
• cograph