Cliques of Orders Three and Four in the Paley-Type Graphs

Abstract

Let \(n=2^s p_{1}^{\alpha _{1}}\cdots p_{k}^{\alpha _{k}}\), where \(s=0\) or 1, \(\alpha _i\ge 1\), and the distinct primes \(p_i\) satisfy \(p_i\equiv 1\pmod {4}\) for all \(i=1, \ldots , k\). Let \(\mathbb {Z}_n^*\) denote the group of units in the commutative ring \(\mathbb {Z}_n\). In a recent paper, we defined the Paley-type graph \(G_n\) of order n as the graph whose vertex set is \(\mathbb {Z}_n\) and xy is an edge if \(x-y\equiv a^2\pmod n\) for some \(a\in \mathbb {Z}_n^*\). Computing the number of cliques of a particular order in a Paley graph or its generalizations has been of considerable interest. In our recent paper, for primes \(p\equiv 1\pmod 4\) and \(\alpha \ge 1\), by evaluating certain character sums, we found the number of cliques of order 3 in \(G_{p^\alpha }\) and expressed the number of cliques of order 4 in \(G_{p^\alpha }\) in terms of Jacobi sums. In this article we give combinatorial proofs and find the number of cliques of orders 3 and 4 in \(G_n\) for all n for which the graph is defined.