On the diameter of the intersection graph of a finite simple group

Czechoslovak Mathematical Journal

Xuanlong Ma¹ 

[1] College of Mathematics and Information Science, Guangxi University, No. 100, Daxue Road, Nanning 530004, Guangxi, People's Republic of China, and Laboratory of Mathematics and Complex Systems, School of Mathematical Sciences, Beijing Normal University, Number 19, XinJieKouWai Street, HaiDian, Beijing 100875, People's Republic of China

关键词: intersection graph;    finite simple group;    diameter;   

DOI  :  

学科分类：数学（综合）

来源: Akademie Ved Ceske Republiky

【 摘 要 】

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 $X\cap Y\ne1$, 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.

【 授权许可】

Unknown