{
  "id": "2207.07214",
  "version": "v1",
  "published": "2022-07-14T21:41:09.000Z",
  "updated": "2022-07-14T21:41:09.000Z",
  "title": "More on minors of Hermitian (quasi-)Laplacian matrix of the second kind for mixed graphs",
  "authors": [
    "Qi Xiong",
    "Gui-Xian Tian",
    "Shu-Yu Cui"
  ],
  "comment": "16 pages,7 figures",
  "categories": [
    "math.CO"
  ],
  "abstract": "A mixed graph $M_{G}$ is the graph obtained from an unoriented simple graph $G$ by giving directions to some edges of $G$, where $G$ is often called the underlying graph of $M_{G}$. In this paper, we introduce two classes of incidence matrices of the second kind of $M_{G}$, and discuss the determinants of these two matrices for rootless mixed trees and unicyclic mixed graphs. Applying these results, we characterize the explicit expressions of various minors for Hermitian (quasi-)Laplacian matrix of the second kind of $M_{G}$. Moreover, we give two sufficient conditions that the absolute values of all the cofactors of Hermitian (quasi-)Laplacian matrix of the second kind are equal to the number of spanning trees of the underlying graph $G$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2022-07-14T21:41:09.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "05C50",
      "15A18"
    ],
    "keywords": [
      "second kind",
      "laplacian matrix",
      "absolute values",
      "unoriented simple graph",
      "sufficient conditions"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 16,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}