Hypergeometric Functions for Dirichlet Characters and Peisert-Like Graphs on $$\mathbb {Z}_n$$

Abstract

For a prime $$p\equiv 3\pmod 4$$ and a positive integer t, let $$q=p^{2t}$$ . The Peisert graph of order q is the graph with vertex set $$\mathbb {F}_q$$ such that ab is an edge if $$a-b\in \langle g^4\rangle \cup g\langle g^4\rangle $$ , where g is a primitive element of $$\mathbb {F}_q$$ . In this paper, we construct a similar graph with vertex set as the commutative ring $$\mathbb {Z}_n$$ 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 $$\mathbb {Z}_n$$ , we consider $$n=p^\alpha $$ or $$2p^\alpha $$ , where $$p\equiv 1\pmod 4$$ is a prime and $$\alpha $$ is a positive integer. For primes $$p\equiv 1\pmod 8$$ , we compute the number of triangles in the graph $$G^*(p^{\alpha })$$ by evaluating certain character sums. Next, we study cliques of order 4 in $$G^*(p^{\alpha })$$ . To find the number of cliques of order 4 in $$G^*(p^{\alpha })$$ , we first introduce hypergeometric functions containing Dirichlet characters as arguments and then express the number of cliques of order 4 in $$G^*(p^{\alpha })$$ in terms of these hypergeometric functions.