The conjugacy classes, conjugate graph and conjugacy class graph of some finite metacyclic 2-groups

In this paper, G denotes a metacyclic 2-group of order at most 32 and ? denotes a simple undirected graph. A conjugacy class is an equivalence relation, in which the group is partitioned into disjoint sets. The conjugate graph is a graph whose vertices are non-central elements of G in which two vertices are adjacent if they are conjugate. The conjugacy class graph is a graph whose vertices are non-central conjugacy classes of a group G in which two vertices are connected if their cardinalities are not coprime. In this paper, the conjugacy classes of metacyclic 2-groups of order at most 32 are computed. The results obtained are then applied to graph theory, more precisely to conjugate graph and conjugacy class graph. Some graph properties such as chromatic number, clique number, dominating number and independent number are also determined