Work

Defective eigenvalues of the non-backtracking matrix

Year: 2025

Type: article

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 are called nontrivial. We develop 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.

Source: Electronic Journal of Linear Algebra

DOI: 10.13001/ela.2025.8835

Authors: Kristin Heysse, Kate Lorenzen, Carolyn Reinhart

Institutions: Macalester College, Swarthmore College

Language: English

Topic: Graph theory and applications

Subfield: Geometry and Topology

Field: Mathematics

Domain: Physical Sciences

Open Access status: diamond

Funders: National Science Foundation, American Institute of Mathematics, National Security Agency, Division of Mathematical Sciences

Awards: G2475220182, Mathematical Sciences Research Institute (MSRI), G329468232, G6019427330, G7845798442