Some Aspects of the Conjugacy Class Graph of the Direct Product of Special Linear Group and K-metacyclic Group

Graphs play a crucial role in studying algebraic structures such as groups, as they help reveal various properties of groups by representing them as graphs and vice versa. In this paper, we aim to explore the conjugacy class graphs associated with the direct product of the special linear group SL(2,q), consisting of 2×2 matrices over a field of order q (where q is an odd prime) and the K-metacyclic group of order p(p-1). We begin by determining the conjugacy classes of the direct product using their respective conjugacy classes and then investigate various graph properties, including planarity, connectivity, chromatic number, independence number, and dominating number.