The Difference Subgroup Graph of a Finite Group

TL;DR

The paper defines and analyzes the Difference Subgroup Graph $D(G)$ and its reduced version $D^*(G)$ for finite groups, establishing that $D(G)$ is edgeless for abelian (and certain Dedekind/Iwasawa-like) groups and linking graph properties to fundamental group-theoretic traits. It develops a broad structural program: connectivity criteria, forbidden-subgraph characterizations, and the interplay between independence/clique numbers and solvability or nilpotency, including universal-vertex phenomena that force a two-prime-order/semidirect-product structure $G\cong \mathbb{Z}_q^\beta \rtimes \mathbb{Z}_{p^\alpha}$. The work shows that bipartiteness forces nilpotency, clawfreeness yields supersolvability, and being a cograph yields solvability; sharp bounds on independence and clique numbers translate into solvability or nilpotency constraints, with detailed considerations of minimal simple groups and classical families. The paper ends by outlining open problems (connectivity of $D^*(G)$, reconstruction from $D(G)$, perfection implying solvability, and potential improvements to clique bounds) that guide future exploration at the interface of graph theory and finite group theory.

Abstract

The \emph{difference subgroup graph} $D(G)$ of a finite group $G$ is defined as the graph whose vertices are the non-trivial proper subgroups of $G$, with two distinct vertices $H$ and $K$ adjacent if and only if $\langle H, K \rangle = G$ but $HK \ne G$. This graph arises naturally as the difference between the join graph $Δ(G)$ and the comaximal subgroup graph $Γ(G)$. In this paper, we initiate a systematic study of $D(G)$ and its reduced version $D^*(G)$, obtained by removing isolated vertices. We establish several fundamental structural properties of these graphs, including conditions for connectivity, forbidden subgraph characterizations, and the relationship between graph parameters - such as independence number, clique number, and girth - and the solvability or nilpotency of the underlying group. The paper concludes with a discussion of open problems and potential directions for future research.