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Let G1 and G2 be simple graphs and let n1 = |V (G1)|, m1 = |E(G1)|, n2 = |V (G2)| and m2 = |E(G2)|. In this paper we derive sharp upper and lower bounds for the number of spanning trees τ in the Cartesian product G1 □G2 of G1 and G2. We show that: and . We also characterize the graphs for which equality holds. As a by-product we derive a formula for the number of spanning trees in Kn1 □Kn2 which turns out to be .

Sharp uppper and lower bounds on the number of spanning trees in Cartesian product graphs