Intersection subgroup graph with forbidden subgraphs

2023/08/22 by Mandal, Santanu, Manna, Pallabi
#05C25 #Combinatorics (math.CO) #FOS: Mathematics

paper · doi:10.48550/arxiv.2308.11301

Abstract

Let G be a group. The intersection subgroup graph of G (introduced by Anderson et al. \citeanderson) is the simple graph Γ_S(G) whose vertices are those non-trivial subgroups say H of G with H∩ K=\e\ for some non-trivial subgroup K of G; two distinct vertices H and K are adjacent if and only if H∩ K=\e\, where e is the identity element of G. In this communication, we explore the groups whose intersection subgroup graph belongs to several significant graph classes including cluster graphs, perfect graphs, cographs, chordal graphs, bipartite graphs, triangle-free and claw-fee graphs. We categorize each nilpotent group G so that Γ_S(G) belongs to the above classes. We entirely classify the simple group of Lie type whose intersection subgroup graph is a cograph. Moreover, we deduce that Γ_S(G) is neither a cograph nor a chordal graph if G is a torsion-free nilpotent group.

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