05C50 Online Seminar: Polona Oblak

The spectral properties of matrices with prescribed patterns have received significant attention in recent research. This talk examines symmetric matrices that share the off-diagonal zero-nonzero pattern with the adjacency matrix of a given graph, focusing on the spectra these matrices can achieve. This problem is widely recognized as the Inverse Eigenvalue Problem for a Graph.

For a simple (not necessarily connected) graph, we introduce the concept of a multiplicity matrix, where each column represents an ordered multiplicity list of eigenvalues realized by a matrix corresponding to its connected component. We present the combinatorial notion of compatibility of two multiplicity matrices and show that their existence guarantees the existence of an orthogonal symmetric matrix corresponding to the join of two graphs. In some cases, this necessary condition is also sufficient, and we present several families of joins of graphs that are realizable by a matrix with only two distinct eigenvalues.

This talk is based on the joint work with Rupert H. Levene and Helena Šmigoc.