Conjugacy Classes and Conjugate Graphs of Frobenius Group of Order 20

The conjugate graph of a finite group G is a graph with vertex set non-central elements of G and two vertices are adjacent if and only if they are conjugate. In contrast, the conjugacy class graph is a graph with vertex set non-central conjugacy classes and two distinct classes are adjacent if and only if their cardinalities are not coprime. In this paper, some mathematical proofs are provided to determine the graph structure of conjugate and conjugacy class graphs associated to a nonabelian metabelian group namely Frobenius group of order 20. In addition, some graph properties of these two types of graphs such as chromatic number, clique number, dominating number and independent number are also found.