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The Sandpile Group of a Cone Over a Bi-Coconut Tree

Dorian Smith

TL;DR

The paper analyzes the spanning-tree number and the sandpile group of cones over bi-coconut trees, providing an explicit formula for the spanning-tree count τ in terms of p, s1, s2 via t(p,s1,s2) and a generating function for t. It then derives the full abelian structure of the sandpile group K(Cone(T(p,s1,s2))) by reducing the reduced Laplacian to a Smith normal form, with a parity- and mod-3 dependent case distinction that yields a ≔ t(p,s1,s2)/2^{s1+s2-2} and explicit decompositions into 2-power factors and a cyclic component Z_a. The authors also prove a corollary demonstrating a family of trees whose cone sandpile groups are cyclic while the number of leaves grows unbounded, addressing a question posed by Reiner and Smith. A separate comb-tree result shows that μ(Cone(T_p)) = 1 and ℓ(T_p) = p, highlighting a sharp contrast between the number of generators and leaf count in certain tree families.

Abstract

The sandpile group of a connected graph is a finite abelian group whose cardinality is the number of spanning trees in the graph. We compute the spanning tree number and sandpile group structure for the cone over a bi-coconut tree, generalizing work of Reiner and Smith on the cone over a coconut tree. We also answer one of their questions, by exhibiting a family of trees whose sandpile groups are all cyclic but their number of leaves grows without bound.

The Sandpile Group of a Cone Over a Bi-Coconut Tree

TL;DR

The paper analyzes the spanning-tree number and the sandpile group of cones over bi-coconut trees, providing an explicit formula for the spanning-tree count τ in terms of p, s1, s2 via t(p,s1,s2) and a generating function for t. It then derives the full abelian structure of the sandpile group K(Cone(T(p,s1,s2))) by reducing the reduced Laplacian to a Smith normal form, with a parity- and mod-3 dependent case distinction that yields a ≔ t(p,s1,s2)/2^{s1+s2-2} and explicit decompositions into 2-power factors and a cyclic component Z_a. The authors also prove a corollary demonstrating a family of trees whose cone sandpile groups are cyclic while the number of leaves grows unbounded, addressing a question posed by Reiner and Smith. A separate comb-tree result shows that μ(Cone(T_p)) = 1 and ℓ(T_p) = p, highlighting a sharp contrast between the number of generators and leaf count in certain tree families.

Abstract

The sandpile group of a connected graph is a finite abelian group whose cardinality is the number of spanning trees in the graph. We compute the spanning tree number and sandpile group structure for the cone over a bi-coconut tree, generalizing work of Reiner and Smith on the cone over a coconut tree. We also answer one of their questions, by exhibiting a family of trees whose sandpile groups are all cyclic but their number of leaves grows without bound.
Paper Structure (13 sections, 9 theorems, 92 equations, 6 figures)

This paper contains 13 sections, 9 theorems, 92 equations, 6 figures.

Table of Contents

  1. Introduction and Preliminaries
  2. Sandpile groups and Laplacians
  3. Main results
  4. Proof of Theorem \ref{['spanning-tree-number-theorem']}
  5. Proof of Theorem \ref{['generating-function-theorem']}
  6. Proof of Theorem \ref{['sandpile-group-theorem']}
  7. Proof outline
  8. Step 1.
  9. Step 2.
  10. Step 3.
  11. Step 4.
  12. Step 5.
  13. Proof of Theorem \ref{['comb-tree-theorem']}

Key Result

Theorem 1.1

For $p \geq 1$ and $s_1,s_2 \geq 1$, the cone over the bi-coconut tree $T(p,s_1,s_2)$ has

Figures (6)

  • Figure :
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  • ...and 1 more figures

Theorems & Definitions (14)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Lemma 1.5
  • Theorem 1.6
  • Theorem 1.7
  • proof
  • proof
  • Lemma 4.1
  • ...and 4 more