Abstract
Let G be a finite group. The intersection graph Δ 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 ∩ Y ≠ 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.
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The research has been supported by NSF of China (11361006), SRF of Guangxi University (XGZ130761) and Yunnan Educational Committee (2014Y500).
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Ma, X. On the diameter of the intersection graph of a finite simple group. Czech Math J 66, 365–370 (2016). https://doi.org/10.1007/s10587-016-0261-2
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DOI: https://doi.org/10.1007/s10587-016-0261-2