Conjugate Graph and Conjugacy Class Graph related to Direct Product of Dihedral Groups

Let 𝐺 be a non-abelian group. The conjugate graph of 𝐺 is a graph whose vertices are the noncentral elements of 𝐺 and two vertices are adjacent if they belong to the same conjugacy class. The conjugacy class graph of 𝐺 is a graph whose vertices are the non-central conjugacy classes of 𝐺 and two vertices 𝑎, 𝑏 are adjacent if 𝑔𝑐𝑑(|𝑎|, |𝑏|) is greater than one. In this paper, we explore the structures of these graphs for the groups 𝐷𝑛 × 𝐷𝑚 for odd and even values of 𝑛 and 𝑚. The chromatic number and independence number of the conjugate graphs, their line graphs and complement graphs are found. We discuss various graph parameters like the existence of Eulerian and Hamiltonian cycles, planarity, connectedness, chromatic number, clique number, independence number, and diameter of the conjugacy class graphs of these groups.