On enumeration of spanning trees of complete multipartite graphs containing a fixed spanning forest

TL;DR

This paper derives a determinantal formula for counting spanning trees in complete multipartite graphs that include a specific spanning forest, extending previous bipartite graph results using algebraic methods.

Contribution

It introduces a new algebraic approach using the Generalized Matrix Determinant Lemma to generalize spanning tree enumeration to complete multipartite graphs.

Findings

Provides a determinantal formula for spanning trees with a fixed forest

Extends known bipartite results to multipartite graphs

Uses algebraic techniques for combinatorial enumeration

Abstract

We present a determinantal formula for the number of spanning trees of a complete multipartite graph containing a given spanning forest $F$. Our approach relies on the Generalized Matrix Determinant Lemma and Jacobi's formula for the derivative of a determinant. This work generalizes known results for complete bipartite graphs and offers an algebraic perspective on the problem.