{
  "id": "1310.1003",
  "version": "v1",
  "published": "2013-10-03T15:04:39.000Z",
  "updated": "2013-10-03T15:04:39.000Z",
  "title": "The signature of line graphs and power trees",
  "authors": [
    "Long Wang",
    "Yi-Zheng Fan"
  ],
  "doi": "10.1016/j.laa.2014.01.020",
  "categories": [
    "math.CO"
  ],
  "abstract": "Let $G$ be a graph and let $A(G)$ be the adjacency matrix of $G$. The signature $s(G)$ of $G$ is the difference between the positive inertia index and the negative inertia index of $A(G)$. Ma et al. [Positive and negative inertia index of a graph, Linear Algebra and its Applications 438(2013)331-341] conjectured that $-c_3(G)\\leq s(G)\\leq c_5(G),$ where $c_3(G)$ and $c_5(G)$ respectively denote the number of cycles in $G$ which have length $4k+3$ and $4k+5$ for some integers $k \\ge 0$, and proved the conjecture holds for trees, unicyclic or bicyclic graphs. It is known that $s(G)=0$ if $G$ is bipartite, and the signature is closely related to the odd cycles or nonbipartiteness of a graph from the existed results. In this paper we show that the conjecture holds for the line graph and power trees.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2013-10-03T15:04:39.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "05C50"
    ],
    "keywords": [
      "line graph",
      "power trees",
      "negative inertia index",
      "conjecture holds",
      "positive inertia index"
    ],
    "tags": [
      "journal article"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2013arXiv1310.1003W"
    }
  }
}