# Brouwer's conjecture holds asymptotically almost surely

## TL;DR

This paper proves that Brouwer's conjecture is almost surely true for large random graphs and extends this result to weighted graphs with negative weights, showing it holds with high probability as the number of vertices grows.

## Contribution

It demonstrates that Brouwer's conjecture holds asymptotically almost surely for large random and weighted graphs, including those with negative weights.

## Key findings

- Brouwer's conjecture holds with probability tending to one as vertices increase.
- The result extends to weighted graphs with negative weights.
- Most graphs with large vertices satisfy Brouwer's conjecture.

## Abstract

We show that for a sequence of random graphs Brouwer's conjecture holds true with probability tending to one as the number of vertices tends to infinity. Surprisingly, it was found that a similar statement holds true for weighted graphs with possible negative weights as well. For graphs with a fixed number of vertices, the result implies that there are constants $C>0$ and $n_{0}$ such that if $n\geq n_{0}$ then among all $2^{{n \choose 2}}$ graphs with $n$ vertices, at least $\left(1-\exp\left(-Cn\right)\right)2^{{n \choose 2}}$ graphs satisfy Brouwer's conjecture.