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

TL;DR

This paper generalizes Paley graphs to finite rings $Z_n$, constructs such graphs, and analyzes their properties, including counting complete subgraphs of sizes 3 and 4, extending known results from finite fields to rings.

Contribution

It introduces a new class of graphs based on $Z_n$, extending Paley graph concepts from finite fields to rings, and computes the number of complete subgraphs within these graphs.

Findings

Count of triangles (3-cliques) in the graph over $Z_{p^{ ext{alpha}}}$.

Count of 4-cliques in the graph over $Z_{p^{ ext{alpha}}}$.

Properties of the constructed graphs related to clique counts.

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 $\mathbb{Z}_{p^{\alpha}}$.