Number of spanning trees containing a given forest

TL;DR

This paper derives a formula for counting spanning trees in a complete graph that contain a specific forest, showing the count depends only on component sizes and total vertices, not the forest's structure.

Contribution

The paper provides a new explicit formula for the number of spanning trees containing a given forest in a complete graph, independent of the forest's internal structure.

Findings

Number of spanning trees containing a given forest depends only on component sizes and total vertices.

Explicit formula:  = q_1 q_2 \u2026 q_m n^{m-2}.

Result is independent of the forest's structure.

Abstract

We consider all spanning trees of a complete simple graph $\Gamma$ on $n$ vertices that contain a given $m-$forest $F$. We show that the number of such spanning trees, $\tau(F)$, doesn't depend on the structure of $F$ and is completely determined by the number of vertices $q_i \, (i=1, ..., m)$ in each connected component of $F$. Specifically, $\tau(F) = q_1 q_2 \cdots q_m n^{m-2}$.