Non-backtracking Spectrum: Unitary Eigenvalues and Diagonalizability

TL;DR

This paper thoroughly analyzes the eigenvalues of the non-backtracking matrix of graphs, focusing on those with magnitude one, their relation to subgraphs, and conditions for diagonalizability, with implications for spectral graph theory.

Contribution

It provides a complete characterization of eigenvalues with magnitude one and proposes a conjecture on the conditions for diagonalizability of the non-backtracking matrix.

Findings

Eigenvalues of magnitude one are characterized and linked to specific subgraphs.

A conjecture on the necessary and sufficient conditions for diagonalizability is formulated.

An interlacing-type result for the Perron eigenvalue is established.

Abstract

Much effort has been spent on characterizing the spectrum of the non-backtracking matrix of certain classes of graphs, with special emphasis on the leading eigenvalue or the second eigenvector. Much less attention has been paid to the eigenvalues of small magnitude; here, we fully characterize the eigenvalues with magnitude equal to one. We relate the multiplicities of such eigenvalues to the existence of specific subgraphs. We formulate a conjecture on necessary and sufficient conditions for the diagonalizability of the non backtracking matrix. As an application, we establish an interlacing-type result for the Perron eigenvalue.