Spectral properties of the non-backtracking matrix of a graph

TL;DR

This paper analyzes the spectral properties of the non-backtracking matrix of a graph, providing methods to compute eigenvectors and eigenvalues, and demonstrating how certain graph properties are determined by its spectrum.

Contribution

It introduces a way to derive eigenvectors from smaller matrices, relates eigenvalues to those of the adjacency matrix, and shows spectrum determines key graph features.

Findings

Eigenvectors of the non-backtracking matrix can be obtained from smaller matrices.

Eigenvalues relate to those of the adjacency matrix, allowing spectral bounds.

Graph properties like components and bipartiteness are spectrum-dependent.

Abstract

We investigate the spectrum of the non-backtracking matrix of a graph. In particular, we show how to obtain eigenvectors of the non-backtracking matrix in terms of eigenvectors of a smaller matrix. Furthermore, we find an expression for the eigenvalues of the non-backtracking matrix in terms of eigenvalues of the adjacency matrix and use this to upper-bound the spectral radius of the non-backtracking matrix and to give a lower bound on the spectrum. We also investigate properties of a graph that can be determined by the spectrum. Specifically, we prove that the number of components, the number of degree 1 vertices, and whether or not the graph is bipartite are all determined by the spectrum of the non-backtracking matrix.