On the roots of the node reliability polynomial

TL;DR

This paper investigates the roots of the node reliability polynomial in graphs, revealing that all such polynomials have nonreal roots, with complex roots densely filling the entire complex plane, contrasting with known properties of all-terminal reliability roots.

Contribution

It proves that node reliability polynomials of connected graphs with at least three nodes always have nonreal roots and that their complex roots are dense in the complex plane, a stark contrast to all-terminal reliability.

Findings

All node reliability polynomials have at least one nonreal root.

The real roots of node reliability polynomials are unbounded.

Complex roots are dense in the entire complex plane.

Abstract

Given a graph $G$ whose edges are perfectly reliable and whose nodes each operate independently with probability $p\in[0,1],$ the node reliability of $G$ is the probability that at least one node is operational and that the operational nodes can all communicate in the subgraph that they induce; it is the analogous node measure of robustness to the well studied \textit{all-terminal reliability}, where the nodes are perfectly reliable but the edges fail randomly. In sharp contrast to what is known about the roots of the all-terminal reliability polynomial, we show that the node reliability polynomial of any connected graph on at least three nodes has a nonreal polynomial root, the collection of real roots of all node reliability polynomials is unbounded, and the collection of complex roots of all node reliability polynomials is dense in the entire complex plane.