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Automated Counting of Spanning Trees for Several Infinite Families of Graphs

Pablo Blanco, Doron Zeilberger

Abstract

Using the theoretical basis developed by Yao and Zeilberger, we consider certain graph families whose structure results in a rational generating function for sequences related to spanning tree enumeration. Said families are Powers of Cycles and Powers of Path; later, we briefly discuss Torus graphs and Grid graphs. In each case we know, a priori, that the set of spanning trees of the family of graphs can be described in terms of a finite-state-machine, and hence there is a finite transfer-matrix that guarantees the generating function is rational. Finding this ``grammar'', and hence the transfer-matrix is very tedious, so a much more efficient approach is to use experimental mathematics. Since computing numerical determinants is so fast, one can use the matrix tree theorem to generate sufficiently many terms, then fit the data to a rational function. The whole procedure can be done rigorously a posteriori.

Automated Counting of Spanning Trees for Several Infinite Families of Graphs

Abstract

Using the theoretical basis developed by Yao and Zeilberger, we consider certain graph families whose structure results in a rational generating function for sequences related to spanning tree enumeration. Said families are Powers of Cycles and Powers of Path; later, we briefly discuss Torus graphs and Grid graphs. In each case we know, a priori, that the set of spanning trees of the family of graphs can be described in terms of a finite-state-machine, and hence there is a finite transfer-matrix that guarantees the generating function is rational. Finding this ``grammar'', and hence the transfer-matrix is very tedious, so a much more efficient approach is to use experimental mathematics. Since computing numerical determinants is so fast, one can use the matrix tree theorem to generate sufficiently many terms, then fit the data to a rational function. The whole procedure can be done rigorously a posteriori.

Paper Structure

This paper contains 16 sections, 25 theorems, 35 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Counting Total Number of Leaves
  4. The B-Z Constant: Average Number of Leaves per Vertex
  5. Powers of a Cycle: Experimental Results
  6. Generating Functions for the Number of Spanning Trees
  7. Generating functions for the Total Number of Leaves
  8. B-Z Constants
  9. Powers of a Path: Experimental Results
  10. Generating functions for the Number of Spanning Trees
  11. Generating functions for the Total Number of Leaves
  12. B-Z Constants
  13. B-Z Constants for Grid and Torus Graphs
  14. Estimating B-Z Constants with Random Sampling
  15. Accompanying Maple package
  16. ...and 1 more sections

Key Result

Theorem 1.0.1

concrete A sequence $(a_0,a_1,\dots)$ is C-finite (of order $r$) with recurrence if and only if for some polynomial $p(x)$ of degree at most $r-1$.

Figures (2)

  • Figure 1: On the left, ${C_7}^{2}$. On the right, ${P_6}^{3}$. The thicker edges represent the edges from the corresponding original graph.
  • Figure 2: On the left, the $3\times 3$ grid graph. On the right, the $3\times 3$ torus graph.

Theorems & Definitions (27)

  • Theorem 1.0.1
  • Proposition 3.0.1
  • Corollary 3.0.2
  • Proposition 4.0.1
  • Proposition 4.0.2
  • Theorem 5.2.1
  • Theorem 5.2.2
  • Theorem 5.2.3
  • Theorem 5.3.1
  • Theorem 5.3.2
  • ...and 17 more