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

TL;DR

This paper establishes precise upper and lower bounds for the number of spanning trees in the Cartesian product of two graphs, providing formulas and characterizations for cases of equality, including a specific formula for complete graphs.

Contribution

It derives sharp bounds and explicit formulas for spanning trees in Cartesian graph products, advancing understanding of their combinatorial properties.

Findings

Derived sharp bounds for spanning trees in Cartesian products.

Provided a formula for complete graph Cartesian products.

Characterized graphs where bounds are tight.

Abstract

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 $\tau$ in the Cartesian product $G_1 \square G_2$ of $G_1$ and $G_2$. We show that: $$ \tau(G_1 \square G_2) \geq \frac{2^{(n_1-1)(n_2-1)}}{n_1n_2} (\tau(G_1) n_1)^{\frac{n_2+1}{2}} (\tau(G_2)n_2)^{\frac{n_1+1}{2}}$$ and $$\tau(G_1 \square G_2) \leq \tau(G_1)\tau(G_2) [\frac{2m_1}{n_1-1} + \frac{2m_2}{n_2-1}]^{(n_1-1)(n_2-1)}.$$ 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 $n_{1}^{n_1-2}n_2^{n_2-2}(n_1+n_2)^{(n_1-1)(n_2-1)}.$