Generalized Sharp Bounds on the Spectral Radius of Digraphs

TL;DR

This paper introduces generalized bounds on the spectral radius of directed graphs, which are tighter and more broadly applicable, especially for sparse and strongly connected digraphs, with conditions for equality.

Contribution

It provides new spectral radius bounds for digraphs, including conditions for equality and applicability to a wider class of graphs than previous bounds.

Findings

Bounds are often tighter than existing ones.

Bounds are particularly suited for sparse digraphs.

Equality conditions relate to outdegree regularity and spectral properties.

Abstract

The spectral radius {\rho}(G) of a digraph G is the maximum modulus of the eigenvalues of its adjacency matrix. We present bounds on {\rho}(G) that are often tighter and are applicable to a larger class of digraphs than previously reported bounds. Calculating the final bound pair is particularly suited to sparse digraphs. For strongly connected digraphs, we derive equality conditions for the bounds, relating to the outdegree regularity of the digraph. We also prove that the bounds hold with equality only if {\rho}(G) is the r-th root of an integer, where r divides the index of imprimitivity of G.