The conjugate graph and conjugacy class graph of order at most 32

Abstract

A groupis called metacyclic if it has a cyclic normal subgroup such that the quotient groupis also cyclic. The classification of non-Abelianmetacyclicp-groups of class two has been found by earlier researcher, which is partitioned into two families of non-isomorphic p-groups. The conjugacy classes of these groups are then applied into graph theory. The conjugate graph is a graph whose the vertices are non-central elements of a finite non-Abelian group. Besides, the conjugacy class graph is a graph whose vertices are non-central of a group that is two vertices are connected if their cardinalities are not coprime, in which their greatest common divisor between the vertices is not equal to one. In this study, the conjugacy classes of the metacyclic 2-groups of order at most 32 have been obtained using the definition of conjugacy classes and their group presentations. The conjugate graph and conjugacy class graph of metacyclic 2-groups of order at most 32 are found directly using the definition. These conjugate graph and conjugacy class graph are then used to determine some graph properties such as chromatic number, clique number, dominating number and independent number. The conjugate graph of the groups turned out to be union of complete components of K2, meanwhile the conjugacy class graph of the groups turned out to be a complete graph.