Geodesic
On the diameter of the intersection graph of a finite simple group
Czechoslovak Mathematical Journal, Tome 66 (2016) no. 2, pp. 365-370

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Let $G$ be a finite group. The intersection graph $\Delta _G$ of $G$ is an undirected graph without loops and multiple edges defined as follows: the vertex set is the set of all proper nontrivial subgroups of $G$, and two distinct vertices $X$ and $Y$ are adjacent if $X\cap Y\ne 1$, where $1$ denotes the trivial subgroup of order $1$. A question was posed by Shen (2010) whether the diameters of intersection graphs of finite non-abelian simple groups have an upper bound. We answer the question and show that the diameters of intersection graphs of finite non-abelian simple groups have an upper bound $28$. In particular, the intersection graph of a finite non-abelian simple group is connected.
DOI : 10.1007/s10587-016-0261-2
Classification : 05C25, 20E32
Keywords: intersection graph; finite simple group; diameter 
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Ma, Xuanlong. On the diameter of the intersection graph of a finite simple group. Czechoslovak Mathematical Journal, Tome 66 (2016) no. 2, pp. 365-370. doi: 10.1007/s10587-016-0261-2

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