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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 $p_i\equiv 1(mod4)$ for all $i=1,\dots ,k$. Let ${\mathbb{Z}}_n^{\ast }$ 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 ...${\mathbb{}}_{}$${}^{\mathrm{}}$${\mathbb{}}_{}^{}$$\mathrm{}\mathrm{}$$\mathrm{}$${}_{{}^{}}$${}_{{}^{}}$${}_{}$

Let D λ d , k denote the family of diagonal hypersurface over a finite field F q given by D λ d , k : X 1 d + X 2 d = λ d X 1 k x 2 d − k , where d ≥ 2, 1 ≤ k ≤ d − 1, and gcd ⁡ ( d , k ) = 1. Let # D λ d , k denote the number of points on D λ d ,...