The diameter of the commuting graph of a finite group with trivial centre

Abstract

The commuting graph Γ of a finite group with trivial centre is examined. It is shown that the connected components of Γ have diameter at most 10.

Perfect commuting graphs

“…Recall that the prime graph of a group G is the graph whose vertices are the primes dividing G, with an edge between distinct vertices p and q if G has a element of order pq. It has been shown by Morgan and Parker [23] that if G is centreless, then .G/ is connected if and only if the prime graph of G is connected.…”

“…This has answered (negatively) an influential conjecture of Iranmanesh and Jafarzadeh [16], that there was a universal upper bound for the diameter of a connected commuting graph. Morgan and Parker [23] have shown, using the classification of finite simple groups, that if Z.G/ is trivial, then the diameter of any connected component of .G/ is at most 10.…”

Maximal non-commuting sets in certain unipotent upper-triangular linear groups

“…Parker, G.L. Morgan and G. Michael (See [15,16,19]). Maximal non-commuting sets in finite groups arise in many contexts in the literature.…”

Spectrum of commuting graphs of some classes of finite groups

“…The commuting graph of a non-abelian group G, denoted by Γ G , is a simple undirected graph whose vertex set is G\Z(G), and two vertices x and y are adjacent if and only if xy = yx. Various aspects of commuting graphs of different finite groups can be found in [3,6,10,11,12,13]. In this paper, we initiate the study of spectrum of commuting graphs of finite nonabelian groups.…”