The present article is designed to be a contribution to the chapter 'Combinatorial Matrix Theory and Graphs' of the Handbook of Linear Algebra, to be published by CRC Press. The format of the handbook is to give just definitions, theorems, and examples; no proofs. In the five sections given...

Let k be a natural number and let G be a graph with at least k vertices. A.E. Brouwer conjectured that the sum of the k largest Laplacian eigenvalues of G is at most e(G) (k choose 2), where e(G) is the number of edges of G. We prove this conjecture for k = 2. We also show that if G is a tree,...