On a Paley-type graph on $\mathbb{Z}_n$

Abstract

Let q be a prime power such that q ≡ 1 (mod 4). The Paley graph of order q is the graph with vertex set as the finite field Fq and edges defined as, ab is an edge if and only if a − b is a non-zero square in Fq. We attempt to construct a similar graph of order n, where n is a positive integer. For suitable n, we construct the graph where the vertex set is the finite commutative ring Zn and edges defined as, ab is an edge if and only if a − b ≡ x 2 (mod n) for some unit x of Zn. We look at some properties of this graph. For primes p ≡ 1 (mod 4), Evans, Pulham and Sheehan computed the number of complete subgraphs of order 4 in the Paley graph of order p. Very recently, Dawsey and McCarthy find the number of complete subgraphs of order 4 in the generalized Paley graph of order q. In this article, we find the number of complete subgraphs of order 3 and 4 in our graph defined over Zn.