# Colloquium abstracts

**Shmuel Friedland**

University of Illinois at Chicago

December 17, 2009

**On the eigenvalues of graphs: results and conjectures**: In this talk we will discuss two topics. First, upper estimates on the maximal eigenvalue, (Perron-Frobenius eigenvalue), of graphs: undirected, bipartite and directed graphs, with prescribed number of vertices and edges. We will characterize in certain cases the graphs which have the biggest maximal eigenvalue.

Second we will discuss the Grone-Merris conjecture for Laplacians of the eigenvalues of graphs. This conjecture states that the eigenvalue sequence of the Laplacian of a given simple undirected graph is majorized by the the dual sequence of the degrees of the graph, and equality holds for threshold graphs.