Speaker
Description
The Nonnegative Inverse Eigenvalue Problem (NIEP) is the problem of characterizing the lists $\sigma$ of $n$ complex numbers (counting multiplicities) that are the spectrum of a nonnegative matrix of size $n$. A list $\sigma$ is said to be realizable if there exists a nonnegative matrix whose spectrum is $\sigma$.
Another way of facing the NIEP is to focus the attention on the characteristic polynomial $P(x)$ of $\sigma$, and to characterize the real monic polynomials of degree $n$ that are the spectrum of a nonnegative matrix of size $n$. Note that a nonnegative matrix $A$ can be seen as the adjacency matrix of a weighted digraph. A real monic polynomial $P(x)$ is said to be realizable if there is a weighted digraph, or equivalently a nonnegative matrix, whose characteristic polynomial is $P(x)$.
Our interest here lies in identifying, among realizable spectra/polynomials, those that are realizable by an irreducible matrix/a strongly connected weighted digraph. We use a special type of digraph structure, called EBL, in which the cycles are especially simple. The EBL digraphs are a tool that allow to relate the information contained in the cyclic structure of the digraph associated to $A$ to the coefficients of its characteristic polynomial $P(x)$, and vice versa, in a suitable way.
After giving some general background, we make some useful new observations and show the existence of irreducible nonnegative and positive realizations in some general cases. We discuss both types of realizations for $n< 5$, where the NIEP is solved. Finally, we study the trace 0 case and, using graph theoretic methods and EBL digraphs, characterize nonnegative irreducible realizability among realizable polynomials.
This talk is based on a joint work with C.R. Johnson and C. Marijuán.