Critical probabilities for site and bond percolation models

.Any infinite graph $G = (V, E)$ has a site percolation critical probability pcˢⁱᵗᵉ and a bond percolation critical probability pcᵇᵒⁿᵈ. The well-known weak inequality pcˢⁱᵗᵉ ≥ pcᵇᵒⁿᵈ is strengthened to strict inequality for a c c broad category of graphs G, including all the usual finite-dimensional lattices in two and more dimensions. The complementary inequality pcˢⁱᵗᵉ ≤ 1 - (1 - pcᵇᵒⁿᵈ)Δ - 1 is proved also, where Δ denotes the supremum of the vertex degrees of G.