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On the number of spanning trees in random regular graphs

Catherine Greenhill, Matthew Kwan, David Wind

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.

On the number of spanning trees in random regular graphs

Abstract

Let be a fixed integer. We give an asympotic formula for the expected number of spanning trees in a uniformly random -regular graph with vertices. (The asymptotics are as , restricted to even if 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) . Numerical evidence is presented which supports our conjecture.

Paper Structure

This paper contains 4 sections, 4 theorems, 10 equations.

Table of Contents

  1. Introduction
  2. Notation and our main results
  3. Plan of attack
  4. Expected number of spanning trees

Key Result

Theorem \oldthetheorem

Let $d\geq 3$ be a fixed integer. Then

Theorems & Definitions (5)

  • Theorem \oldthetheorem
  • Theorem \oldthetheorem
  • Conjecture \oldthetheorem
  • Theorem \oldthetheorem
  • Lemma \oldthetheorem