On a Paley-Type Graph on $${\mathbb {Z}}_n$$

Abstract

Let q be a prime power such that \(q\equiv 1\pmod {4}\). The Paley graph of order q is the graph with vertex set as the finite field \({\mathbb {F}}_q\) and edges defined as, ab is an edge if and only if \(a-b\) is a non-zero square in \({\mathbb {F}}_q\). We attempt to construct a similar graph of order n, where \(n\in {\mathbb {N}}\). For suitable n, we construct the graph where the vertex set is the finite commutative ring \({\mathbb {Z}}_n\) and edges defined as, ab is an edge if and only if \(a-b\equiv x^2\pmod {n}\) for some unit x of \({\mathbb {Z}}_n\). We look at some properties of this graph. For primes \(p\equiv 1\pmod {4}\), Evans, Pulham and Sheehan computed the number of complete subgraphs of order 4 in the Paley graph. 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, for primes \(p\equiv 1\pmod {4}\) and any positive integer \(\alpha\), we find the number of complete subgraphs of order 3 and 4 in our graph defined over \({\mathbb {Z}}_{p^{\alpha }}\).