Spanning Trees in Grid Graphs

Paul Raff
Rutgers University
praff@math.rutgers.edu

Introduction

I undertook a project to more clearly understand the structure of the sequence of the number of spanning trees of G × Pn, where n ranges from 1 to infinity. I created an enumeration scheme that counted these spanning trees and many other related objects. Full details of my method can be found in my paper, which has been submitted to the Journal of Integer Sequences. This webpage contains the results for small graphs. Click on any graph below to see all of the results related to that graph.

This project could not have been achieved without the help of Mathematica, including the proper presentation of results.

If you would like to find out more about this project or if you would like to become part of the extension, then feel free to contact me.

$\mathrm{ShowGraph}\left[\mathrm{GraphProduct}\left[\mathrm{PathGraph}\left[5\right],\mathrm{ToMathematicaGraph}\left[G,2\right]\right]\right]$

$\left(\begin{array}{cc}3& 1\\ 2& 1\end{array}\right)$

$\left(\text{⁠}\begin{array}{c}1\\ 1\end{array}\text{⁠}\right)$

$x^2-4x+1$

$\mathrm{Factor}\left[\mathrm{char}\right]//\mathrm{TraditionalForm}$

$x^2-4x+1$

$\mathrm{GeneratingFunctionFromPolynomialAndSequence}\left[\mathrm{rec},\mathrm{seq}\right]//\mathrm{TraditionalForm}$

$\frac{x}{x^2-4x+1}$

Created by Wolfram Mathematica 6.0  (06 March 2009)