Hypergeometric functions for Dirichlet characters and Peisert-like graphs

Abstract

For a prime p ≡ 3 (mod 4) and a positive integer t, let q = p 2t . The Peisert graph of order q is the graph with vertex set Fq such that ab is an edge if a − b ∈ g 4 ∪ g g 4 , where g is a primitive element of Fq. In this paper, we construct a similar graph with vertex set as the commutative ring Zn for suitable n, which we call Peisert-like graph and denote by G(n). Owing to the need for cyclicity of the group of units of Zn, we consider n = p α or 2p α , where p ≡ 1 (mod 4) is a prime and α is a positive integer. For primes p ≡ 1 (mod 8), we compute the number of triangles in the graph G(p α ) by evaluating certain character sums. Next, we study cliques of order 4 in G(p α ). To find the number of 4-order cliques in G(p α ), we first introduce hypergeometric functions containing Dirichlet characters as arguments, and then express the number of 4-order cliques in G(p α ) in terms of these hypergeometric functions.