Automated Counting of Spanning Trees for Several Infinite Families of Graphs

TL;DR

This paper presents an efficient experimental mathematics approach to derive rational generating functions for spanning trees in specific infinite graph families, leveraging the matrix tree theorem and finite-state machine representations.

Contribution

It introduces a practical method combining numerical computation and data fitting to find spanning tree generating functions for complex graph families.

Findings

Generated rational functions for Powers of Cycles and Paths

Demonstrated efficiency of experimental mathematics approach

Discussed extensions to Torus and Grid 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.