Let G be a group. By using a family َA of subsets of automorphisms of G, we introduced a simple graph T_A(G), which is a generalization of the non-commuting graph. In this paper, we study the combinatorial properties of T_A(G).

Abstract. If the centre of a group G is trivial, then so is the centre of its automorphism group. We study the structure of the centre of the automorphism group of a group G when the centre of G is a cyclic group.

In particular...

the

clique number and automorphism group of n

R, where R = Zp^2 or R = Zp^3 ....

The aim of this paper is to identify complete power graphs of groups and compute their clique number and show that power graphs are perfect. Moreover, automorphism of the power graph of cyclic groups is determined here.

We classify planar graphs and complete power graphs of groups and show that the only infinite group

with a complete power graph is the Prüfer group Z_{p^∞}. Clique and chromatic numbers and the automorphism group of

power graphs...