Written: April 2025

Maple package

• ExpSP.txt, A maple package for computing generating functions enumerating the number of spanning trees in numerous infinite familes of graphs.

Sample Input and Output for ExpSP.txt

• If you want to see the generating functions (that are always rational functions), in n, for the the number of spanning trees of the graphs H_n,r defined in the paper, for 2 ≤ r ≤ 5, as well as generating functions for the "sum of the number of leaves" over all spanning trees (that enables) computing the exact average number of leaves, as well as precise asymptotics
  the input gives the output

  If you want to see JUST the generating functions (that are always rational functions), in n, for the the number of spanning trees of the graphs H_n,r defined in the paper, for 2 ≤ r ≤ 7,
  the input gives the output

  ---

• If you want to see the generating functions (that are always rational functions), in n, for the the number of spanning trees of the graphs G_n,r defined in the paper, for 2 ≤ r ≤ 5, as well as generating functions for the "sum of the number of leaves" over all spanning trees (that enables) computing the exact average number of leaves, as well as precise asymptotics
  the input gives the output

  ---

• If you want to see the generating functions (that are always rational functions), in n, for the the number of spanning trees of the r by n rectangular grid, for 2 ≤ r ≤ 4, as well as generating functions for the "sum of the number of leaves" over all spanning trees (that enables) computing the exact average number of leaves, as well as precise asymptotics
  the input gives the output

• If you want to see JUST the generating functions (that are always rational functions), in n, for the the number of spanning trees of the r by n rectangular grid, for 2 ≤ r ≤ 6
  the input gives the output

  ---

• If you want to see the generating functions (that are always rational functions), in n, for the the number of spanning trees of the r by n rectangular torus, for 2 ≤ r ≤ 6, as well as generating functions for the "sum of the number of leaves" over all spanning trees (that enables) computing the exact average number of leaves, as well as precise asymptotics
  the input gives the output

  ---

• If you want to see estimated statistics about the number of leaves in a random spanning tree of H_n,r for various n and several r
  the input gives the output

• If you want to see estimated statistics about the number of leaves in a random spanning tree of G_n,r for various n and several r
  the input gives the output

• If you want to see estimated statistics about the number of leaves in a random labeled tree on n vertices for various n
  the input gives the output

• If you want to see a comparison of the exact average of the number of leaves in a random spanning trees for our four favorite families,
  See here

  ---