Hypergeometric functions for Dirichlet characters and Peisert-like graphs on ${\mathbb{Z}}_n$

For a prime $p\equiv 3(\mathrm{m}\mathrm{o}\mathrm{d}4)$ 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 $a-b\in ⟨g^4⟩\cup g⟨g^4⟩$, 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 $\mathit{\text{Peisert-like}}$ graph and denote by $G^{\ast }(n)$. Owing to the need for cyclicity of the group of units of ${\mathbb{Z}}_n$, we consider $n=p^{\alpha }$ or $2p^{\alpha }$, where $p\equiv 1(\mathrm{m}\mathrm{o}\mathrm{d}4)$ is a prime and $\alpha$ is a positive integer. For primes $p\equiv 1(\mathrm{m}\mathrm{o}\mathrm{d}8)$, we compute the number of triangles in the graph $G^{\ast }(p^{\alpha })$ by evaluating certain character sums. Next, we study cliques of order 4 in $G^{\ast }(p^{\alpha })$. To find the number of cliques of order $4$ in $G^{\ast }(p^{\alpha })$, we first introduce hypergeometric functions containing Dirichlet characters as arguments, and then express the number of cliques of order $4$ in $G^{\ast }(p^{\alpha })$ in terms of these hypergeometric functions.