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

Wei Wang, Jun Ge

TL;DR

This paper addresses counting spanning trees of a complete multipartite graph $K_{n_1,\ldots,n_s}$ that contain a fixed spanning forest $F$. It introduces a determinant-based approach by modeling the contracted graph Laplacian $L(K_{n_1,\ldots,n_s}/F)$ as a rank-$s$ perturbation of a diagonal matrix, and applying the Generalized Matrix Determinant Lemma and Jacobi's formula to derive a compact formula for $\tau_F(K_{n_1,\ldots,n_s})$. The main contribution is a determinantal expression in terms of $\alpha_p$, $n_{ip}$, and a small $s\times s$ cofactor matrix, which generalizes Moon-type results and recovers the known $s=2$ formula of Dong and Ge. This algebraic framework provides a principled method for counting spanning trees with a fixed forest in complete multipartite graphs and connects to the Matrix Tree Theorem through determinant derivatives.

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.

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

TL;DR

This paper addresses counting spanning trees of a complete multipartite graph that contain a fixed spanning forest . It introduces a determinant-based approach by modeling the contracted graph Laplacian as a rank- perturbation of a diagonal matrix, and applying the Generalized Matrix Determinant Lemma and Jacobi's formula to derive a compact formula for . The main contribution is a determinantal expression in terms of , , and a small cofactor matrix, which generalizes Moon-type results and recovers the known formula of Dong and Ge. This algebraic framework provides a principled method for counting spanning trees with a fixed forest in complete multipartite graphs and connects to the Matrix Tree Theorem through determinant derivatives.

Abstract

We present a determinantal formula for the number of spanning trees of a complete multipartite graph containing a given spanning forest . 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.
Paper Structure (4 sections, 15 theorems, 47 equations)

This paper contains 4 sections, 15 theorems, 47 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. The structure of $L(K_{n_1,\ldots,n_s}/F)$
  4. Main result

Key Result

Lemma 2.1

Every cofactor of $L(G)$ is equal to the number of spanning trees of $G$, that is, $\operatorname{adj} L(G)=\tau(G)J$.

Theorems & Definitions (26)

  • Lemma 2.1: Matrix-Tree Theorem kirchhoff1847biggs1993
  • Corollary 2.2
  • Corollary 2.3
  • Lemma 2.4: Generalized Matrix Determinant Lemma harville2008
  • Lemma 2.5: Jacobi's formula harville2008
  • Lemma 3.1
  • proof
  • Proposition 3.2
  • proof
  • Proposition 3.3
  • ...and 16 more