***This event is in-person and open only to Princeton University ID holders**

# Old and New Results on the Spread of the Spectrum of a Graph

The spread of a matrix is defined as the diameter of its spectrum. This quantity has been well-studied for general matrices and has recently grown in popularity for the specific case of the adjacency matrix of a graph. Most notably, Gregory, Herkowitz, and Kirkland proved a number of key results for the spread of a graph and made two key conjectures regarding graphs that maximize spread. In particular, they conjectured that the maximum spread over all graphs with a fixed number of vertices is given by the join of a clique and independent set and that the maximum spread over all graphs with a fixed number of vertices and edges is given by a bipartite graph if one exists. In this talk, I will review some of the key results regarding the spread of a general matrix, some known results for the specific case of an adjacency matrix, and give a high-level outline of a recent paper regarding these two conjectures. This is joint work with Jane Breen, Alex Riasanovsky, and Michael Tait.

*John Urschel is a member of the Institute for Advanced Study. Urschel recently completed his Ph.D. in math at MIT, under the supervision of Michel Goemans. His research interests include numerical analysis, graph theory, and data science/machine learning.*