A formula for the number of spanning trees in circulant graphs with non-fixed generators and discrete tori

TL;DR

This paper derives a simplified formula for counting spanning trees in circulant graphs with linearly dependent generators and applies similar methods to discrete tori, providing insights into their spanning tree entropy.

Contribution

It introduces a new formula with fewer factors for counting spanning trees in circulant graphs with variable generators, improving upon the matrix tree theorem.

Findings

Derived a formula with ewer factors for circulant graphs

Compared spanning tree entropy between fixed and non-fixed generators

Extended methods to discrete tori spanning trees

Abstract

We consider the number of spanning trees in circulant graphs of $\beta n$ vertices with generators depending linearly on $n$. The matrix tree theorem gives a closed formula of $\beta n$ factors, while we derive a formula of $\beta-1$ factors. Using the same trick, we also derive a formula for the number of spanning trees in discrete tori. Moreover, the spanning tree entropy of circulant graphs with fixed and non-fixed generators is compared.