Proof of Brouwers Conjecture (BC) for all graphs with number of vertices n > n_0 assuming that BC holds for n< n_0 for some n_0

TL;DR

This paper proves Brouwers Conjecture for all graphs with more than n_0 vertices, assuming it holds for smaller graphs, thus extending the conjecture's validity to larger graphs.

Contribution

It establishes the validity of Brouwers Conjecture for all sufficiently large graphs based on its validity for smaller graphs.

Findings

Brouwers Conjecture holds for all n > n_0 if it holds for n < n_0

Provides a method to extend conjecture validity from smaller to larger graphs

Strengthens the theoretical understanding of Laplacian eigenvalues in graph theory

Abstract

Assuming that Brouwers Conjecture the upper bound for the sum of t< n largest eigenvalues of Laplacian graph on n vertices true for n <n_0, we prove the Brouwers Conjecture BC for n > n_0 for some fixed n_0