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Matrix Quasi-tree Theorem

Qingying Deng, Xian'an Jin, Qi Yan, Yexiang Yan

TL;DR

The paper generalizes the Matrix Tree Theorem to all embedded graphs by introducing symbolic skew-adjacency matrices and a reduction map that convert determinants into polynomials encoding spanning quasi-trees. The central result, the Matrix Quasi-tree Theorem, shows that $f(\det(I_n + \mathbf{A_{\\sigma}^s})) \\bmod 2$ equals the edge-sets of spanning quasi-trees, enabling the exact counting of quasi-trees via polynomial evaluation. The authors develop the framework of ribbon graphs, partial duality, and delta-matroids to extend the result from bouquets to general connected ribbon graphs, providing a complete topological analogue to Kirchhoff's theorem for orientable and non-orientable embeddings. This establishes new algebraic tools for quasi-tree enumeration and links to delta-matroid theory and topological graph concepts.

Abstract

Building on prior work that established Matrix Quasi-tree Theorems for special embedded graphs, in this paper, we develop a comprehensive theory applicable to all embedded graphs. We introduce symbolic skew-adjacency matrices and reduction maps as key innovations, and prove that a specific polynomial derived from these matrices encodes all spanning quasi-trees of a bouquet. This result provides a complete analogue of the Matrix Tree Theorem for topological graph theory, with applications to quasi-tree enumeration in both orientable and non-orientable embedded graphs.

Matrix Quasi-tree Theorem

TL;DR

The paper generalizes the Matrix Tree Theorem to all embedded graphs by introducing symbolic skew-adjacency matrices and a reduction map that convert determinants into polynomials encoding spanning quasi-trees. The central result, the Matrix Quasi-tree Theorem, shows that equals the edge-sets of spanning quasi-trees, enabling the exact counting of quasi-trees via polynomial evaluation. The authors develop the framework of ribbon graphs, partial duality, and delta-matroids to extend the result from bouquets to general connected ribbon graphs, providing a complete topological analogue to Kirchhoff's theorem for orientable and non-orientable embeddings. This establishes new algebraic tools for quasi-tree enumeration and links to delta-matroid theory and topological graph concepts.

Abstract

Building on prior work that established Matrix Quasi-tree Theorems for special embedded graphs, in this paper, we develop a comprehensive theory applicable to all embedded graphs. We introduce symbolic skew-adjacency matrices and reduction maps as key innovations, and prove that a specific polynomial derived from these matrices encodes all spanning quasi-trees of a bouquet. This result provides a complete analogue of the Matrix Tree Theorem for topological graph theory, with applications to quasi-tree enumeration in both orientable and non-orientable embedded graphs.

Paper Structure

This paper contains 4 sections, 8 theorems, 44 equations, 1 figure, 1 table.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Symbolic skew-adjacency matrices
  4. Proof of the Matrix Quasi-tree Theorem

Key Result

Theorem 1.1

Let $B$ be an orientable bouquet with edge set $[n]= \{1, 2, \dots, n\}$ and signed rotation $\sigma$, and let $\mathbf{A_{\sigma}^u}$ be its unsymbolic skew-adjacency matrix. Then the number of spanning quasi-trees of $B$ is

Figures (1)

  • Figure 1: A signed rotation of the bouquet is $[-1^a, 2^a, 3^a, 1^b, 2^b, -4^a,3^b,-5^a,4^b,5^b]$.

Theorems & Definitions (22)

  • Theorem 1.1: Bouchet871997LauriMacrisMerino2023
  • Theorem 1.3: Matrix Quasi-tree Theorem
  • Example 1.4
  • Definition 2.1: bollobas
  • Definition 2.2: ChunJCTA
  • Proposition 2.3: Chun
  • Definition 3.1
  • Remark 3.2
  • Definition 3.3
  • Remark 3.4
  • ...and 12 more