# Seymour's second-neighborhood conjecture from a different perspective

## TL;DR

This paper explores Seymour's Second-Neighborhood Conjecture by reformulating it using linear algebra, offering a new perspective on this longstanding graph theory problem.

## Contribution

It introduces alternative linear algebraic formulations of Seymour's conjecture, providing a novel approach to understanding and potentially proving the conjecture.

## Key findings

- New linear algebraic statements of the conjecture
- Potential pathways for algebraic proof strategies
- Insight into the structure of directed graphs related to the conjecture

## Abstract

Seymour's Second-Neighborhood Conjecture states that every directed graph whose underlying graph is simple has at least one vertex $v$ such that the number of vertices of out-distance $2$ from $v$ is at least as large as the number of vertices of out-distance $1$ from it. We present alternative statements of the conjecture in the language of linear algebra.