# On the DLW Conjectures

## TL;DR

This paper proves Conjecture A related to polynomial properties over finite fields, which were originally conjectured in 2007 and recently confirmed for Conjecture 2, strengthening understanding of permutation polynomials and monomial graphs.

## Contribution

It provides a proof of Conjecture A, a strengthened form of a conjecture about permutation polynomials over finite fields, advancing the theoretical understanding of these structures.

## Key findings

- Conjecture 2 and 1 have been confirmed.
- The paper proves Conjecture A, a strengthening of Conjecture 2.
- Results contribute to the theory of permutation polynomials and monomial graphs.

## Abstract

In 2007, Dmytrenko, Lazebnik and Williford posed two related conjectures about polynomials over finite fields. Conjecture~1 is a claim about the uniqueness of certain monomial graphs. Conjecture~2, which implies Conjecture~1, deals with certain permutation polynomials of finite fields. Two natural strengthenings of Conjecture~2, referred to as Conjectures~A and B in the present paper, were also insinuated. In a recent development, Conjecture~2 and hence Conjecture~1 have been confirmed. The present paper gives a proof of Conjecture~A.