On the two-sided group digraph with a normal adjacency matrix

This article explores the adjacency matrix of a two-sided group graph andits properties. We introduce the two-sided color group digraph to generalize the Cayley color graph and the two-sided group digraph. We alsoobtain the adjacency matrix of the latter digraph and provide a criterion fordetermining the normality of the adjacency matrix of a two-sided group graph.Moreover, we prove that if all the two-sided group digraphs of valency two fora certain group G are normal, then G is a Hamiltonian group. We also showthat if a strongly connected two-sided group digraph of valency two is normal,the corresponding group is isomorphic to the product of two groups: a cyclicgroup with either Tk,n or Hp,q, or an abelian group.