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

Let $G_1$ and $G_2$ be simple graphs and let $n_1 = |V(G_1)|$, $m_1 = |E(G_1)|$, $n_2 = |V(G_2)|$ and $m_2 = |E(G_2)|.$ In this paper we derive sharp upper and lower bounds for the number of spanning trees $τ$ in the Cartesian product $G_1 \square G_2$ of $G_1$ and $G_2$. We show that: $$ τ(G_1 \square G_2) \geq \frac{2^{(n_1-1)(n_2-1)}}{n_1n_2} (τ(G_1) n_1)^{\frac{n_2+1}{2}} (τ(G_2)n_2)^{\frac{n_1+1}{2}}$$ and $$τ(G_1 \square G_2) \leq τ(G_1)τ(G_2) [\frac{2m_1}{n_1-1} + \frac{2m_2}{n_2-1}]^{(n_1-1)(n_2-1)}.$$ We also characterize the graphs for which equality holds. As a by-product we derive a formula for the number of spanning trees in $K_{n_1} \square K_{n_2}$ which turns out to be $n_{1}^{n_1-2}n_2^{n_2-2}(n_1+n_2)^{(n_1-1)(n_2-1)}.$