On diagonal equations over finite fields via walks in NEPS of graphs

In this paper, we obtain an explicit combinatorial formula for the number of solutions $(x_1,\dots ,x_r)\in {\mathbb{F}}_{p^{ab}}$ to the diagonal equation $x_1^k+\cdots +x_r^k=\alpha$ over the finite field ${\mathbb{F}}_{p^{ab}}$, with $k=\frac{p^{ab}-1}{b(p^a-1)}$ and $b>1$ by using the number of $r$-walks in NEPS of complete graphs.