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arXiv · 2403.04157

On the diameter of intersection graphs of finite groups

Melissa Lee·Kamilla Rekvényi
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Abstract

The intersection graph $\Delta_G$ of a finite group $G$ is a simple graph with vertices the non-trivial proper subgroups of $G$, and an edge between two vertices if their corresponding subgroups intersect non-trivially. These graphs were introduced by Cs\'ak\'any and Poll\'ak in 1969. In this paper we answer two long-standing open questions posed by Cs\'ak\'any and Poll\'ak concerning the diameter of intersection graphs. We prove some necessary conditions for a non-simple group to have an intersection graph of diameter 4. We also construct the first examples of non-simple groups and alternating groups whose intersection graphs have diameter 4.

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Melissa Lee, Kamilla Rekvényi. 2024-03-07. On the diameter of intersection graphs of finite groups. https://arxiv.org/abs/2403.04157

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