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

Nowroozi Larki, Farzaneh; Rayat Pisheh, Shahram

doi:10.22061/jdma.2024.11041.1072

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

Document Type : Full Length Article

Authors

¹ Mathematics, Basic Faculty, Shahid Rajaee Teacher Training University

² Department of Mathematics, Faculty of Science, Shahid Rajaee Teacher Training University

10.22061/jdma.2024.11041.1072

Abstract

This article explores the adjacency matrix of a two-sided group graph and
its properties. We introduce the two-sided color group digraph to generalize the Cayley color graph and the two-sided group digraph. We also
obtain the adjacency matrix of the latter digraph and provide a criterion for
determining 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 for
a certain group G are normal, then G is a Hamiltonian group. We also show
that 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 cyclic
group with either Tk,n or Hp,q, or an abelian group.

Graphical Abstract