ON COEXIST CENTRALIZER GRAPH OF THE DIRECT PRODUCT OF DIHEDRAL GROUP AND GROUP OF UNITS

ON COEXIST CENTRALIZER GRAPH

Abstract

 Coexist centralizer graph of finite groups is defined in this paper, it is a graph of a finite group $G$ with vertex set as the non-central elements of $G$ in such away that two distinct elements are adjacent in the graph if and only if they belong to the same centralizer. The work has to do with the centralizer graph of the direct product of the dihedral group and the group of units. It is shown that the graph is not connected, chromatic number and the clique number of the coexist centralizer graph of the direct product of the dihedral group and the group of units are equal, a relationship between the clique number and the center of the group under consideration is also established. Illustrations were done for certain orders of the groups.