{
  "id": "2006.00002",
  "version": "v1",
  "published": "2020-05-29T10:39:27.000Z",
  "updated": "2020-05-29T10:39:27.000Z",
  "title": "No signed graph with the nullity $η(G,σ)=|V(G)|-2m(G)+2c(G)-1$",
  "authors": [
    "Yong Lu",
    "Jingwen Wu"
  ],
  "comment": "15pages, 2 figures",
  "categories": [
    "math.CO"
  ],
  "abstract": "Let $G^{\\sigma}=(G,\\sigma)$ be a signed graph and $A(G,\\sigma)$ be its adjacency matrix. Denote by $m(G)$ the matching number of $G$. Let $\\eta(G,\\sigma)$ be the nullity of $(G,\\sigma)$. He et al. [Bounds for the matching number and cyclomatic number of a signed graph in terms of rank, Linear Algebra Appl. 572 (2019), 273--291] proved that $$|V(G)|-2m(G)-c(G)\\leq\\eta(G,\\sigma)\\leq |V(G)|-2m(G)+2c(G),$$ where $c(G)$ is the dimension of cycle space of $G$. Signed graphs reaching the lower bound or the upper bound are respectively characterized by the same paper. In this paper, we will prove that no signed graphs with nullity $|V(G)|-2m(G)+2c(G)-1$. We also prove that there are infinite signed graphs with nullity $|V(G)|-2m(G)+2c(G)-s,~(0\\leq s\\leq3c(G), s\\neq1)$ for a given $c(G)$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2020-05-29T10:39:27.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "05C35",
      "05C50"
    ],
    "keywords": [
      "linear algebra appl",
      "matching number",
      "infinite signed graphs",
      "cyclomatic number",
      "adjacency matrix"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 15,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}