Geometric multiplicity of unitary non-backtracking eigenvalues

TL;DR

This paper characterizes when complex unitary numbers are eigenvalues of the non-backtracking matrix of an undirected graph, providing a formula and an efficient linear-time algorithm to compute their geometric multiplicity.

Contribution

It offers a complete characterization and a novel linear-time algorithm for computing the geometric multiplicity of unitary eigenvalues in non-backtracking matrices.

Findings

Derived a closed-form formula for geometric multiplicity.

Developed a linear-time algorithm for multiplicity computation.

Provided conditions for unitary eigenvalues of non-backtracking matrices.

Abstract

We completely characterize the conditions under which a complex unitary number is an eigenvalue of the non-backtracking matrix of an undirected graph. Further, we provide a closed formula to compute its geometric multiplicity and describe an algorithm to compute this multiplicity without making a single matrix computation. The algorithm has time complexity that is linear in the size of the graph.