For quasi-transitive d-regular graphs, only trees have the minimal percolation threshold 1/(d-1), but without quasi-transitivity cyclic d-regular graphs with this threshold exist.
Strict monotonicity of percolation thresholds under covering maps
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Critical threshold for regular graphs
For quasi-transitive d-regular graphs, only trees have the minimal percolation threshold 1/(d-1), but without quasi-transitivity cyclic d-regular graphs with this threshold exist.