On the quadratic unitary Cayley graphs

Abstract

Quadratic unitary Cayley graphs are a generalization of the well-known Paley graphs. Let Zn be the ring of integers modulo n. The quadratic unitary Cayley graph of Zn, denoted by GZn, is the graph whose vertices are given by the elements of Zn and two vertices u,v∈Zn are adjacent if and only if u−v or v−u is a quadratic unit in Zn. When p≥3 is a prime and ν≥1 is an integer, all the eigenvalues of GZpν have been given in [8]. In this paper, we improve the above result and obtain all the exact eigenvalues of GZ2ν by a new approach. We also determine all the eigenvalues of GZn for general n>1. As an application, we characterize necessary and sufficient conditions on n such that GZn is strongly regular.