Quadratic unitary Cayley graphs are a generalization of the well-known Paley graphs. Let Z(n) be the ring of integers modulo n. The quadratic unitary Cayley graph of Z(n), denoted by G(Zn), is the graph whose vertices are given by the elements of Z(n) and two vertices u, v is an element of Z(n) are adjacent if and only if u-v or v-u is a quadratic unit in Z(n). When p >= 3 is a prime and nu >= 1 is an integer, all the eigenvalues of G(Zp nu) have been given in [8]. In this paper, we improve the above result and obtain all the exact eigenvalues of G(Z2n) by a new approach. We also determine all the eigenvalues of G(Zn) for general n > 1. As an application, we characterize necessary and sufficient conditions on n such that G(Zn) is strongly regular. (c) 2022 Elsevier Inc. All rights reserved.
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