The Perron non-backtracking eigenvalue after node addition

TL;DR

This paper investigates how the Perron eigenvalue of the non-backtracking matrix of a graph changes when a new node is added, establishing that it never decreases and providing rigorous bounds for its variation.

Contribution

It proves an interlacing-type result for the Perron eigenvalue after node addition and introduces the first rigorous bounds for this eigenvalue's change.

Findings

Perron eigenvalue never decreases after node addition

Established bounds for eigenvalue difference

Results depend on diagonalizability assumption

Abstract

Consider a finite undirected unweighted graph G and add a new node to it arbitrarily connecting it to pre-existing nodes. We study the behavior of the Perron eigenvalue of the non-backtracking matrix of G before and after such a node addition. We prove an interlacing-type result for said eigenvalue, namely, the Perron eigenvalue never decreases after node addition. Furthermore, our methods lead to bounds for the difference between the eigenvalue before and after node addition. These are the first known bounds that have been established in full rigor. Our results depend on the assumption of diagonalizability of the non-backtracking matrix. Practical experience says that this assumption is fairly mild in many families of graphs, though necessary and sufficient conditions for it remain an open question.