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

Denis E. Videla

Published 2019-07-06Version 1

In this paper, we find a formula for the number of $r$-walks on NEPS of arbitrary graphs in any basis. We apply this formula to find the number of elements of a commutative ring that can be represented as a sum of $r$ units. We then also obtain the number of solutions $(x_i)_{i=1}^r$ with $x_i\neq 0$ to the diagonal equation $x_{1}^k+\cdots+x_{r}^k=b$ over a finite field $\mathbb{F}_{p^m}$, obtaining an identity for generalized Jacobi sums.