Matrix Quasi-tree Theorem
Combinatorics
2025-12-02 v1
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.
Cite
@article{arxiv.2512.00680,
title = {Matrix Quasi-tree Theorem},
author = {Qingying Deng and Xian'an Jin and Qi Yan and Yexiang Yan},
journal= {arXiv preprint arXiv:2512.00680},
year = {2025}
}
Comments
14 pages, 2 figures