Note on structural properties of graphs

In this paper, we establish sufficient and necessary conditions for the existence of abelian subgroups of maximal order of a finite group G, by means of its commuting graph. The order of these subgroups attains the bound c = vertical bar[x(1)]vertical bar + ... + vertical bar[x(m)]vertical bar, where [x(i)] denotes the conjugacy class of x(i) in G and m is the smallest integer j such that vertical bar[x(1)]vertical bar+ ... + vertical bar[x;]vertical bar >= vertical bar C-G(x(j))vertical bar, where C-G(x(j)) is the centralizer of x(j) in G.

Automated summary

In this paper, we establish sufficient and necessary conditions for the existence of abelian subgroups of maximal order in a finite group G, using its commuting graph. The order of these subgroups is bounded by the sum of the cardinalities of the conjugacy classes and the centralizer of certain elements in G.