Interlacing families and the Hermitian spectral norm of digraphs

TL;DR

This paper proves that for any finite connected graph, an orientation exists that bounds the Hermitian spectral radius by the universal cover's spectral radius, resolving a problem by Mohar using interlacing families of polynomials.

Contribution

It introduces a novel application of interlacing families to Hermitian adjacency matrices, establishing spectral bounds related to graph orientations.

Findings

Existence of orientations with spectral radius bounded by the universal cover

Characterization of equality cases as trees

Application of interlacing polynomial methods to Hermitian matrices

Abstract

It is proved that for any finite connected graph $G$, there exists an orientation of $G$ such that the spectral radius of the corresponding Hermitian adjacency matrix is smaller or equal to the spectral radius of the universal cover of $G$ (with equality if and only if $G$ is a tree). This resolves a problem proposed by Mohar. The proof uses the method of interlacing families of polynomials that was developed by Marcus, Spielman, and Srivastava in their seminal work on the existence of infinite families of Ramanujan graphs.