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

## TL;DR

This paper derives an explicit combinatorial formula for counting solutions to certain diagonal equations over finite fields by analyzing walks in NEPS of graphs, connecting algebraic solutions with graph-theoretic structures.

## Contribution

It introduces a novel combinatorial approach linking solutions of diagonal equations over finite fields to walks in NEPS of complete graphs.

## Key findings

- Explicit formula for solutions to diagonal equations over finite fields.
- Connection established between algebraic equations and graph walks.
- Method applicable to equations with specific parameters.

## Abstract

In this paper, we obtain an explicit combinatorial formula for the number of solutions $(x_1,\ldots,x_r)\in \mathbb{F}_{p^{ab}}$ 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.