JournalsrsmupVol. 131pp. 281–292

A graph related to the join of subgroups of a finite group

  • Hadi Ahmadi

    Isfahan University of Technology, Iran
  • Bijan Taeri

    Isfahan University of Technology, Iran
A graph related to the join of subgroups of a finite group cover
Download PDF

Abstract

For a finite group GG different from a cyclic group of prime power order, we introduce an undirected simple graph Δ(G){\Delta} (G) whose vertices are the proper subgroups of GG which are not contained in the Frattini subgroup of GG and two vertices HH and KK are joined by an edge if and only if G=H,KG\kern -1pt =\kern -1pt \langle H, K\rangle . In this paper we study Δ(G){\Delta} (G) and show that it is connected and determine the clique and chromatic number of Δ(G){\Delta} (G) and obtain bounds for its diameter and girth. We classify finite groups with complete graphs and also classify finite groups with domination number 1. Also we show that if the independence number of the graph Δ(G){\Delta} (G) is at most 7, then GG is solvable.

Cite this article

Hadi Ahmadi, Bijan Taeri, A graph related to the join of subgroups of a finite group. Rend. Sem. Mat. Univ. Padova 131 (2014), pp. 281–292

DOI 10.4171/RSMUP/131-17