Paley-type graphs of order a product of two distinct primes

In this paper, we initiate the study of Paley-type graphs $\Gamma_N$ modulo $N=pq$, where $p$, $q$ are distinct primes of the form $4k+1$. It is shown that $\Gamma_N$ is an edge-regular, symmetric, Eulerian and Hamiltonian graph. Also, the vertex connectivity, edge connectivity, diameter and girth of $\Gamma_N$ are studied and their relationship with the forms of $p$ and $q$ are discussed. Moreover, we specify the forms of primes for which $\Gamma_N$ is triangulated or triangle-free and provide some bounds (exact values in some particular cases) for the order of the automorphism group $\operatorname{Aut}(\Gamma_N)$ of the graph $\Gamma_N$, the chromatic number, the independence number, and the domination number of $\Gamma_N$.