Intersection graph of cyclic subgroups of groups

2015/09/15 by R. Rajkumar, Rajkumar, R., P. Devi +1
Computer Science · Engineering · Mathematics · #05C10 #05C17 #05C25 #Advanced Graph Theory Research #FOS: Mathematics #Finite Group Theory Research #Group Theory (math.GR) #graph theory and CDMA systems #math.GR #msc:05C10 #msc:05C17 #msc:05C25

paper · pdf · doi:10.48550/arxiv.1509.04574

10 pages

arxiv created 2015/09/15 · openalex publication_date 2015/09/15 · arxiv updated 2015/09/16 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Let G be a group. The intersection graph of cyclic subgroups of G, denoted by \mathscr I_c(G), is a graph having all the proper cyclic subgroups of G as its vertices and two distinct vertices in \mathscr I_c(G) are adjacent if and only if their intersection is non-trivial. In this paper, we classify the finite groups whose intersection graph of cyclic subgroups is one of totally disconnected, complete, star, path, cycle. We show that for a given finite group G, girth(\mathscr I_c (G)) ∈ \3, ∞\. Moreover, we classify all finite non-cyclic abelian groups whose intersection graph of cyclic subgroups is planar. Also for any group G, we determine the independence number, clique cover number of \mathscr I_c (G) and show that \mathscr I_c (G) is weakly α-perfect. Among the other results, we determine the values of n for which \mathscr I_c (ℤ_n) is regular and estimate its domination number.

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