Tight lower bound for percolation threshold on a quasi-regular graph
  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. 
Kathleen E. Hamilton and Leonid P. Pryadko
math-ph cond-mat.stat-mech math.MP math.PR