**Financial support through the following projects:**

UIDB/00297/2020

UID/MAT/00297/2019

UID/MAT/00297/2013

PEst-OE/MAT/UI0297/2014

PEst-OE/MAT/UI0297/2011

Wednesday, 5 February 2014, 2:00 p.m.

Lecturer: Professor Charles Johnson (Department of Mathematics, College of William and Mary, Williamsburg, Virginia, US)

Title: "Hollow Symmetric Nonnegative Matrices".

Local: Room 1.4, Edifício VII

Faculdade de Ciências e Tecnologia, Quinta da Torre, Caparica

Abstract: An n-by-n matrix is called "hollow" if all its diagonal entries are 0. Examples of hollow, symmetric, nonnegative (HSN) matrices include 1) distance matrices (various metrics), 2) adjacency matrices of undirected graphs, and 3) matrices of interest in the nonnegative inverse eigenvalue problem. We were initially motivated to study HSN matrices, in looking at special inequalities relating the eigenvalues and diagonal entries of LaPlacians of graphs. The distribution of positive and negative eigenvalues of adjacency matrices plays a big role in understanding such inequalities. However, these matrices are quite interesting on their own. We survey what we have found, including eigenvalue structure and some remarkable Schur complement, cycle structure.