On the full Brouwer's Laplacian spectrum conjecture.

Let G be a simple connected graph and let S k (G) be the sum of the first k largest Laplacian eigenvalues of G. It was conjectured by Brouwer in 2006 that S k (G) ≤ e (G) + ( k + 1 2 ) holds for 1 ≤ k ≤ n − 1. The case k = 2 was proved by Haemers, Mohammadian and Tayfeh-Rezaie [Linear Algebra Appl., 2010]. In this paper, we propose the full Brouwer's Laplacian spectrum conjecture and we prove the conjecture holds for k = 2 which also confirm the conjecture of Guan et al. in 2014. [ABSTRACT FROM AUTHOR]