Geodesic
On the diameter of the intersection graph of a finite simple group
Czechoslovak Mathematical Journal, Tome 66 (2016) no. 2, pp. 365-370 Cet article a éte moissonné depuis la source Czech Digital Mathematics Library

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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.
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

[1] Bosák, J.: The graphs of semigroups. Theory of Graphs and Its Applications Proc. Sympos., Smolenice 1963 Publ. House Czechoslovak Acad. Sci., Praha (1964), 119-125. | MR

[2] Chigira, N., Iiyori, N., Yamaki, H.: Non-abelian Sylow subgroups of finite groups of even order. Invent. Math. 139 (2000), 525-539. | DOI | MR | Zbl

[3] Csákány, B., Pollák, G.: The graph of subgroups of a finite group. Czech. Math. J. 19 (1969), 241-247 Russian. | MR

[4] Herzog, M., Longobardi, P., Maj, M.: On a graph related to the maximal subgroups of a group. Bull. Aust. Math. Soc. 81 (2010), 317-328. | DOI | MR | Zbl

[5] Huppert, B.: Endliche Gruppen I. Die Grundlehren der mathematischen Wissenschaften in Einzeldarstellungen 134 Springer, Berlin German (1967). | MR | Zbl

[6] Kayacan, S., Yaraneri, E.: Abelian groups with isomorphic intersection graphs. Acta Math. Hungar. 146 (2015), 107-127. | DOI | MR | Zbl

[7] Kleidman, P. B., Wilson, R. A.: The maximal subgroups of $J_4$. Proc. Lond. Math. Soc., III. Ser. 56 (1988), 484-510. | DOI | MR

[8] Kondraťev, A. S.: Prime graph components of finite simple groups. Math. USSR Sb. 67 (1990), Russian 235-247. | DOI | MR | Zbl

[9] Sawabe, M.: A note on finite simple groups with abelian Sylow $p$-subgroups. Tokyo J. Math. 30 (2007), 293-304. | DOI | MR | Zbl

[10] Shen, R.: Intersection graphs of subgroups of finite groups. Czech. Math. J. 60 (2010), 945-950. | DOI | MR | Zbl

[11] Williams, J. S.: Prime graph components of finite groups. J. Algebra 69 (1981), 487-513. | DOI | MR | Zbl

[12] Zavarnitsine, A. V.: Finite groups with a five-component prime graph. Sib. Math. J. 54 (2013), 40-46. | DOI | MR | Zbl

[13] Zelinka, B.: Intersection graphs of finite abelian groups. Czech. Math. J. 25 (1975), 171-174. | MR | Zbl

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