Toroidality and projective-planarity of intersection graphs of subgroups of finite groups

Abstract

Let G be a group. The intersection graph of subgroups of G, denoted by I (G), is a graph with all the proper subgroups of G as its vertices and two distinct vertices in I (G) are adjacent if and only if the corresponding subgroups having a non-trivial intersection in G. In this paper, we classify the finite groups whose intersection graph of subgroups are toroidal or projective-planar. In addition, we classify the finite groups whose intersection graph of subgroups are one of bipartite, complete bipartite, tree, star graph, unicyclic, acyclic, cycle, path or totally disconnected. Also we classify the finite groups whose intersection graph of subgroups does not contain one of K

Intersection graphs of cyclic subgroups of groups

“…Remark 2.4. In [13], Zelinka proved that for any group G, α(I (G)) = m, where m is the number of prime order proper subgroups of G. In [9], the authors showed that Θ(I (G)) = α(I (G)). It is interesting to note that by Theorems 2.9 and 2.10, we have α(I (G)) = α(I c (G)) and Θ(I (G)) = Θ(I c (G)).…”

“…Inspired by these, there are several papers appeared in the literature which have studied the intersecting graphs on algebraic structures, viz., rings and modules. See, for instance [1,3,9,10,12] and the references therein.…”

Power graphs of (non)orientable genus two

“…Chiang-Hsieh et al [7] characterized the commutative rings of genus one. Very recently, Rajkumar and Devi [20] classified the finite groups whose intersection graphs of subgroups have (non)orientable genus one. Afkhami et al [2] classified planar, toroidal, and projective commuting and noncommuting graphs of finite groups.…”

Finite groups with star-free noncyclic graphs

“…Akhlaghi and Tong-Viet [4] studied the finite groups with K 4 -free prime graphs. Rajkumar and Devi [15] classified the finite groups with K 4 or K 5 -free intersection graphs of subgroups. In [8], Das and Nongsiang classified K 3 -free commuting graphs of finite non-abelian groups.…”