On the Brouwer-type Conjecture for Signless Laplacian Eigenvalues of Graphs

Abstract

Motivated by Brouwer's conjecture, Ashraf, Omidi and Tayfeh-Rezaie proposed the following Brouwer-type conjecture that for every graph $G$ on $n$ vertices with $m$ edges, the sum $S_k^{+}(G)$ of its $k$ largest signless Laplacian eigenvalues satisfies $S_k^{+}(G)\le m+\left(\genfrac{}{}{0ex}{}{k+1}{2}\right)$ for $k=1,\dots ,n$. In this paper, we prove that the above conjecture holds. Moreover, the equality holds if and only if $k=1$ and $G$ is either star $K_{1,a}$ or triangle $K_3$ with adding some isolated vertices. For split graphs, properties of block signless Laplacian matrices based on clique and independent set are adapted. While for non-split graphs, some spectral graph substructure are used to control the sum of signless Laplacian eigenvalues.

arXiv comment29 pages