Structures and Topological Indices of Commutting Involution Graphs

For a group G and a conjugacy class X of involutions in G, C ( G , X ) is the commuting involution graph with vertex set X and two distinct vertices are adjacent if and only if they commute in G. In this paper, we first review some important results on the disks sizes and diameter of such graphs and then we prove that C ( S z ( q ) , X ) consists of q 2 + 1 cliques each of size q − 1 , where Sz(q) denotes the Suzuki group. We also investigate the structure of C ( G , X ) for some finite simple groups with a strongly embedded subgroups. Some distances or disk related indices and polynomials of such graphs are also computed.