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

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.

Spectral characterization of matchings in graphs

“…In this section we prove that that these eigenvalues can be any k distinct nonzero purely imaginary numbers and their conjugate pairs. Similar to approaches in [3,4] we are going to use the Jacobian method, so we need to define an appropriate function and show its Jacobian is nonsingular when it is evaluated at some point. Let G be a graph on n vertices with matching number k, and k + m edges where m > 0.…”

“…We begin by introducing some required terminology as given in [3]. Let A = [a ij ] be an n × n real skew-symmetric matrix.…”

“…To study matchings in connected graphs we first study matchings in trees. A certain kind of trees called NEB trees is introduced in [3] and it has been shown that any NEB tree has a full matching. We introduce required definitions and notation for NEB trees as given in [3].…”

“…Proof. The forward direction is proved in [3,Observation 3.8]. For the backward direction, assume T is not NEB with respect to any vertex.…”