A note on a conjecture concerning rank one perturbations of singular M-matrices

TL;DR

This paper investigates a conjecture about the eigenvalues of rank one perturbations of singular M-matrices, demonstrating its validity in low dimensions and providing counterexamples in higher dimensions.

Contribution

It disproves a conjecture in dimensions four and higher and confirms its validity in dimension two and under certain conditions in dimension three.

Findings

Conjecture is false in dimension four and above.

Conjecture holds in dimension two.

Conjecture is valid in dimension three with additional conditions.

Abstract

A conjecture from a paper by J. Bierkens and A.C.M. Ran concerning the location of eigenvalues of rank one perturbations of singular M-matrices is shown to be false in dimension four and higher, but true for dimension two, as well as for dimension three with an additional condition on the perturbation.