A partial proof of the Brouwer's conjecture

TL;DR

This paper provides partial proofs for Brouwer's conjecture on the sum of the largest Laplacian eigenvalues in simple graphs, establishing its validity under specific conditions related to graph parameters.

Contribution

The paper offers the first partial proof of Brouwer's conjecture for certain classes of simple graphs with specific bounds on parameters.

Findings

Proves Brouwer's conjecture for graphs with n ≤ m ≤ ((√3−1)/4)(n−1)n.

Validates the conjecture for graphs where k falls within a specific interval based on m and n.

Establishes the conjecture's validity for graphs with k in a defined range related to m and n.

Abstract

Let $G$ be a simple graph with $n$ vertices and $m$ edges and let $k$ be a natural number such that $k\leq n.$ Brouwer conjectured that the sum of the $k$ largest Laplacian eigenvalues of $G$ is at most $m+{k+1 \choose 2}.$ In this paper we prove that this conjecture is true for simple $(m,n)$-graphs where $n\leq m\leq \frac{\sqrt{3}-1}{4}(n-1)n$ and $k\in \left[ \sqrt[3]{\frac{8m^{2}}{n-1}+4mn+n^{2}}, n\right].$ Moreover, we prove that the conjecture is true for all simple $(m,n)$-graphs where $k (\leq n)$ is a natural number from the interval $\left[\sqrt{2n-2m+2\sqrt{2m^{2}+mn(n-1)}},1+\frac{8m^{2}}{n^{2}(n-1)}+\frac{4m}{n}\right].$