On the number of spanning trees in random regular graphs

Abstract: Let $d \geq 3$ be a fixed integer. We give an asympotic formula for the expected number of spanning trees in a uniformly random $d$-regular graph with $n$ vertices. (The asymptotics are as $n\to\infty$, restricted to even $n$ if $d$ is odd.) We also obtain the asymptotic distribution of the number of spanning trees in a uniformly random cubic graph, and conjecture that the corresponding result holds for arbitrary (fixed) $d$. Numerical evidence is presented which supports our conjecture.

• The paper derives an asymptotic expression for the expected number of spanning trees in d-regular graphs as the number of vertices grows.
• It employs the small subgraph conditioning method to reveal the asymptotic distribution, particularly in cubic graphs, highlighting a mix of analytic and probabilistic techniques.
• Numerical simulations back the conjectured distribution model, suggesting significant theoretical implications and practical applications in network reliability.

Analysis of "On the number of spanning trees in random regular graphs" (1309.6710)

The paper "On the number of spanning trees in random regular graphs" presents a comprehensive analysis of calculating the asymptotic number of spanning trees in $d$-regular graphs, where each vertex has the same degree $d$. The authors provide asymptotic expressions for the expected number of spanning trees in uniformly random $d$-regular graphs and discuss the distribution of these counts in specific cases.

Asymptotic Formula for Uniformly Random $d$-Regular Graphs

The authors derive an asymptotic expression to estimate the expected number of spanning trees, denoted $E(Y_{\mathcal{G}})$, in a random $d$-regular graph as the number of vertices $n$ approaches infinity. They find that:

$E Y_{\mathcal{G}} \sim \exp\left(\frac{6d^2 - 14d + 7}{4(d-1)^2}\right) \cdot \frac{(d-1)^{1/2}n(d-2)^{3/2}}{(\frac{(d-1)^{d-1}(d^2-2d)^{d/2-1})^n}.$

This formula captures both a constant scaling factor and a dominant exponential component dependent on $n$. The formula extends results by McKay and others by providing the explicit computation of the constant factor $c_d$.

Distribution Analysis with Small Subgraph Conditioning Method

The distribution of spanning trees in random regular graphs was further analyzed using the small subgraph conditioning method. This method revealed the asymptotic distribution for cubic graphs ($d=3$) and is conjectured to hold for all fixed $d \geq 3$. The asymptotic distribution is expressed in terms of Poisson random variables perturbing around the expected counts, demonstrating concentration near the expected value.

The study defines parameters $\lambda_j(d)$ and $\zeta_j(d)$ which help characterize these distributions. For specific cases such as cubic graphs, it proposes a specific distribution that coincides with a product of independent Poisson-distributed random variables adjusted by combinatorial terms.

Numerical Evidence and Conjectures

Numerical simulations support the conjecture that analogous results of asymptotic distributional forms apply to $d$-regular graphs for arbitrary $d \geq 3$. The empirical results show convergence metrics such as $p_d(n)$ for various $d$, supporting the theoretical conjectures about distribution forms.

Practical and Theoretical Implications

The practical implications of these results impact several domains, such as network reliability (in electrical networks), and provide insights into properties related to the Tutte polynomial and Merino-Welsh conjecture. The theoretical implications extend our understanding of the combinatorial structure and complexity of random graphs and serve as a foundation for further exploration in related areas such as graph homomorphism or spanning tree enumeration models.

Conclusion

The findings of the paper provide a significant contribution to the understanding of spanning tree enumeration in random regular graphs. By delivering precise asymptotic formulas and suggesting distribution models that hold across a spectrum of regular graph configurations, the study invites verification through further empirical and theoretical investigations. The combination of analytic methods and probabilistic tools establishes a robust groundwork for advancing graph theory in both pure mathematics and applied computational fields.