Geodesic
Some results on the main supergraph of~finite~groups
Algebra and discrete mathematics, Tome 30 (2020) no. 2, pp. 172-178.

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Let $G$ be a finite group. The main supergraph $\mathcal{S}(G)$ is a graph with vertex set $G$ in which two vertices $x$ and $y$ are adjacent if and only if $o(x) \mid o(y)$ or $o(y)\mid o(x)$. In this paper, we will show that $G\cong \mathrm{PSL}(2,p)$ or $\mathrm{PGL}(2,p)$ if and only if $\mathcal{S}(G)\cong \mathcal{S}(\mathrm{PSL}(2,p))$ or $\mathcal{S}(\mathrm{PGL}(2,p))$, respectively. Also, we will show that if $M$ is a sporadic simple group, then $G\cong M$ if only if $\mathcal{S}(G)\cong \mathcal{S}(M)$.
Keywords: graph, main supergraph, finite groups, Thompson's problem. 
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A. K. Asboei; S. S. Salehi. Some results on the main supergraph of~finite~groups. Algebra and discrete mathematics, Tome 30 (2020) no. 2, pp. 172-178. http://geodesic.mathdoc.fr/item/ADM_2020_30_2_a1/

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