Talk information

Date: Sunday, November 9, 2025
Time: 10:10–11:00
Place: Schreiber 309
Speaker: Alan Lew (Technion)
Title: Sums of Laplacian eigenvalues and sums of degrees

Abstract:

The Laplacian matrix of a graph, introduced by Kirchhoff in 1847, has since become one of the central objects of study in spectral graph theory. The Laplacian eigenvalues of a graph are closely related to various graph properties, such as the number of spanning trees, connectivity and expansion parameters, the matching number, and even the topology of the clique complex of the graph. In this talk, I will focus on the relation between the Laplacian spectrum and the degree sequence of a graph. In particular, I will discuss a recent result stating that the sum of the $k$ largest Laplacian eigenvalues of a graph is bounded from above by the sum of its $2k$ largest degrees. This bound is tight, and it extends a classical result of Anderson and Morley, corresponding to the case k=1. As an application, I will show how this result leads to new progress towards a conjecture of Brouwer relating the sum of the $k$ largest Laplacian eigenvalues to the number of edges in the graph.