Parameters of Quadratic Residue Digraphs over Certain Finite Fields

Abstract

Linking graph theory and algebra has been a rich area of mathematical exploration for a long time. Cayley digraphs and Zero-Divisor graphs are two such examples. In this paper, we make another connection by constructing and studying digraphs whose vertices are the elements of the multiplicative group of the finite fields \\(\\mathbb{Z}_{p}\\) for certain primes \\(p\\). In particular, we determine parameters, including the diameter of such digraphs and the eccentricity of certain vertices of these digraphs. We also find some results on the quadratic residues and nonresidues of \\(\\mathbb{Z}_{p}\\).