{
  "id": "1402.4326",
  "version": "v1",
  "published": "2014-02-18T13:32:25.000Z",
  "updated": "2014-02-18T13:32:25.000Z",
  "title": "On the inertia set of a signed graph with loops",
  "authors": [
    "Marina Arav",
    "Hein van der Holst",
    "John Sinkovic"
  ],
  "comment": "14 pages, 1 figure",
  "categories": [
    "math.CO"
  ],
  "abstract": "A signed graph is a pair $(G,\\Sigma)$, where $G=(V,E)$ is a graph (in which parallel edges and loops are permitted) with $V=\\{1,\\ldots,n\\}$ and $\\Sigma\\subseteq E$. The edges in $\\Sigma$ are called odd edges and the other edges of $E$ even. By $S(G,\\Sigma)$ we denote the set of all symmetric $n\\times n$ real matrices $A=[a_{i,j}]$ such that if $a_{i,j} < 0$, then there must be an even edge connecting $i$ and $j$; if $a_{i,j} > 0$, then there must be an odd edge connecting $i$ and $j$; and if $a_{i,j} = 0$, then either there must be an odd edge and an even edge connecting $i$ and $j$, or there are no edges connecting $i$ and $j$. (Here we allow $i=j$.) For a symmetric real matrix $A$, the partial inertia of $A$ is the pair $(p,q)$, where $p$ and $q$ are the number of positive and negative eigenvalues of $A$, respectively. If $(G,\\Sigma)$ is a signed graph, we define the \\emph{inertia set} of $(G,\\Sigma)$ as the set of the partial inertias of all matrices $A \\in S(G,\\Sigma)$. In this paper, we present a formula that allows us to obtain the minimal elements of the inertia set of $(G,\\Sigma)$ in case $(G,\\Sigma)$ has a $1$-separation using the inertia sets of certain signed graphs associated to the $1$-separation.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2014-02-18T13:32:25.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "05C22",
      "05C50",
      "15A03"
    ],
    "keywords": [
      "signed graph",
      "inertia set",
      "odd edge",
      "edge connecting",
      "partial inertia"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 14,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2014arXiv1402.4326A"
    }
  }
}