Defective eigenvalues of the non-backtracking matrix

TL;DR

This paper investigates the occurrence of defective eigenvalues and Jordan blocks in the non-backtracking matrix of graphs, providing classifications and constructions for graphs with such spectral properties.

Contribution

It offers a complete classification of graphs with defective eigenvalues in the non-backtracking matrix and constructs infinite graph families exhibiting these properties.

Findings

Identifies conditions for defective eigenvalues in non-backtracking matrices

Classifies eigenspaces relationships for graphs with at most one cycle

Constructs infinite graph families with nontrivial Jordan blocks

Abstract

We consider graphs for which the non-backtracking matrix has defective eigenvalues, or graphs for which the matrix does not have a full set of eigenvectors. The existence of these values results in Jordan blocks of size greater than one, which we call nontrivial. We show a relationship between the eigenspaces of the non-backtracking matrix and the eigenspaces of a smaller matrix, completely classifying their differences among graphs with at most one cycle. Finally, we provide several constructions of infinite graph families that have nontrivial Jordan blocks for both this smaller matrix and the non-backtracking matrix.