Tight lower bound for percolation threshold on a quasi-regular graph

TL;DR

This paper derives an exact expression for the site percolation threshold on quasi-regular trees and provides a tighter lower bound for quasi-regular graphs using the spectral radius of the Hashimoto matrix, advancing percolation theory.

Contribution

It introduces a novel exact formula for percolation thresholds on quasi-regular structures and improves existing bounds with spectral radius techniques.

Findings

Exact expression for p_c on quasi-regular trees

A tighter lower bound for quasi-regular graphs

Bound surpasses inverse spectral radius of the original graph

Abstract

We construct an exact expression for the site percolation threshold p_c on a quasi-regular tree, and a related exact lower bound for a quasi-regular graph. Both are given by the inverse spectral radius of the appropriate Hashimoto matrix used to count non-backtracking walks. The obtained bound always exceeds the inverse spectral radius of the original graph, and it is also generally tighter than the existing bound in terms of the maximum degree.