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.