The nonnegative inverse eigenvalue problem from the coefficients of the characteristic polynomial. EBL digraphs (Q996331)

The nonnegative inverse eigenvalue problem (NIEP) is defined as follows: ``Given a family of complex numbers \(\sigma=\{\lambda_1, \lambda_2, \dots, \lambda_n\}\), find the necessary and sufficient conditions for the existence of a nonnegative matrix \(A\) of order \(n\) with spectrum \(\sigma\)''. The NIEP is trivial for \(n\leq 2\). For the case \(n=3\), and the case \(n=4\) when the spectrum \(\sigma\) is real, \textit{R. Loewy} and \textit{D. London} [``A note on an inverse problem for nonnegative matrices'', Linear Multilinear Algebra 6, 83--90 (1978; Zbl 0376.15006)] solved the NIEP. In the article under review, the authors notice a nonnegative matrix as the adjacency matrix of a weighted digraph. They concentrate on the coefficients of its characteristic polynomial instead of focusing directly on its spectrum. Thus, the NIEP that they consider can be described as follows: ``Given real numbers \(k_1, k_2, \dots, k_n\), find necessary and sufficient conditions for the existence of a nonnegative matrix \(A\) of order \(n\) with characteristic polynomial \(x^n+k_1x^{n-1}+k_2x^{n-2}+\cdots+k_n\)''. The coefficients of the characteristic polynomial are closely related to the cyclic structure of the weighted digraph with adjacency matrix \(A\). They introduce a special type of digraph structure and they call it EBL. Through presenting certain results, they demonstrate the interest of EBL structures. Then, they completely solve the NIEP from the coefficients of the characteristic polynomial for \(n=4\). At the end, they solve a special case of the NIEP for \(n\leq 2p+1\) with \(k_1=k_2=\ldots=k_{p-1}=0\) and \(p\geq 2\). Also, the referee of this article mentions the NIEP for \(n=4\) has been studied by \textit{M. E. Meehan} in her Ph.D. thesis [Some results on matrix spectra, Ph.D. thesis, National University of Ireland Dublin (1998)] with a different approach from the one in this article.