The Difference Subgroup Graph of a Finite Group

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 $\Delta(G)$ and the comaximal subgroup graph $\Gamma(G)$. 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.