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

• Volume: 66, Issue: 2, page 365-370
• ISSN: 0011-4642

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## Abstract

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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.

## How to cite

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Ma, Xuanlong. "On the diameter of the intersection graph of a finite simple group." Czechoslovak Mathematical Journal 66.2 (2016): 365-370. <http://eudml.org/doc/280105>.

@article{Ma2016,
abstract = {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.},
author = {Ma, Xuanlong},
journal = {Czechoslovak Mathematical Journal},
keywords = {intersection graph; finite simple group; diameter},
language = {eng},
number = {2},
pages = {365-370},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {On the diameter of the intersection graph of a finite simple group},
url = {http://eudml.org/doc/280105},
volume = {66},
year = {2016},
}

TY - JOUR
AU - Ma, Xuanlong
TI - On the diameter of the intersection graph of a finite simple group
JO - Czechoslovak Mathematical Journal
PY - 2016
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 66
IS - 2
SP - 365
EP - 370
AB - 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.
LA - eng
KW - intersection graph; finite simple group; diameter
UR - http://eudml.org/doc/280105
ER -

## References

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