Conjugacy classes and conjugate graph of a K-metacyclic group

Abstract

The relationship between algebraic structures and graphs has become an interesting topic of research nowadays. In this paper, we have considered the conjugate graph related to the conjugacy relation of a group. The vertices of the said graph are the noncentral elements of the group, and two vertices are adjacent if they are conjugate. For this particular study, we focused on the conjugate graph of a K-metacyclic group of order p(p-1). We first determine the conjugacy classes of this group and then obtain its conjugate graph. Various graph properties such as planarity, line graph, complement graph, clique number, dominating number, spectrum, and Laplacian are also studied in this paper.