Construction of real skew-symmetric matrices from interlaced spectral data and graph

TL;DR

This paper extends classical spectral graph results to real skew-symmetric matrices, establishing conditions under which such matrices can be constructed with prescribed interlaced eigenvalues for trees and their supergraphs.

Contribution

It introduces new methods for constructing real skew-symmetric matrices with specified spectral properties related to trees and their supergraphs, expanding prior symmetric matrix results.

Findings

Constructed skew-symmetric matrices with prescribed interlaced eigenvalues.

Extended classical spectral results from symmetric to skew-symmetric matrices.

Applicable to certain families of trees and their supergraphs.

Abstract

A 1989 result of Duarte asserts that for a given tree T on n vertices, a fixed vertex i, and two sets of distinct real numbers L, M of sizes n and n-1, respectively, such that M strictly interlaces L, there is a real symmetric matrix A such that graph of A is T, eigenvalues of A are given by L, and eigenvalues of A(i) are given by M. In 2013, a similar result for connected graphs was published by Hassani Monfared and Shader, using the Jacobian method. Analogues of these results are presented here for real skew-symmetric matrices whose graphs belong to a certain family of trees, and all of their supergraphs.