# Formulas counting spanning trees in line graphs and their extensions

## TL;DR

This paper derives formulas for counting spanning trees in graphs and their line and middle graphs, based on specific subgraph conditions, revealing new factorization properties related to graph cuts.

## Contribution

It introduces a novel formula for counting spanning trees containing a given edge set in graphs with particular subgraph structures, extending to line and middle graphs.

## Key findings

- Derived a formula for $ au_G(M)$ under specific conditions.
- Obtained a new factorization property for spanning trees related to graph cuts.
- Applied results to line and middle graphs for spanning tree enumeration.

## Abstract

For any connected multigraph $G=(V,E)$ and any $M\subseteq E$, if $M$ induces an acyclic subgraph of $G$ and removing all edges in $M$ yields a subgraph of $G$ whose components are complete graphs, a formula for $\tau_G(M)$ is obtained, where $\tau_G(M)$ is the number of spanning trees in $G$ which contain all edges in $M$. Applying this result, we can easily obtain a formula for the number of spanning trees in the line graph or the middle graph of an arbitrary graph. Applying this result, we also show that for any connected graph $G$ with a clique $U$ which is a cut-set of $G$, the number of spanning trees in $G$ has a factorization which is analogous to a property of the chromatic polynomial of $G$.