Brouwer's conjecture for the sum of the k largest Laplacian eigenvalues of some graphs

Abstract

Let GG be a graph with n(G)n\left(G) vertices and e(G)e\left(G) edges, and Sk(G){S}_{k}\left(G) be the sum of the kk largest Laplacian eigenvalues of GG. Brouwer conjectured that Sk(G)≤e(G)+k+12{S}_{k}\left(G)\le e\left(G)+\left(\phantom{\rule[-0.75em]{}{0ex}},\genfrac{}{}{0.0pt}{}{k+1}{2}\right) for 1≤k≤n(G)1\le k\le n\left(G). In this article, we obtain upper bounds of Sk(G){S}_{k}\left(G) in terms of the graphs that contain the friendship graph or the book graph as a subgraph. Further, we show that Brouwer’s conjecture holds for Halin graphs and certain classes of graphs.

Keywords

• brouwer’s conjecture
• laplacian eigenvalues
• halin graph
• 05c50
• 05c07
• 05c09