{
  "id": "1408.3303",
  "version": "v2",
  "published": "2014-08-14T14:45:36.000Z",
  "updated": "2015-02-15T02:31:57.000Z",
  "title": "On the spectral radius of a class of non-odd-bipartite even uniform hypergraphs",
  "authors": [
    "Murad-ul-Islam Khan",
    "Yi-Zheng Fan"
  ],
  "categories": [
    "math.CO"
  ],
  "abstract": "In order to investigate the non-odd-bipartiteness of even uniform hypergraphs, starting from a simple graph $G$, we construct a generalized power of $G$, denoted by $G^{k,s}$, which is obtained from $G$ by blowing up each vertex into a $k$-set and each edge into a $(k-2s)$-set, where $s \\le k/2$. When $s < k/2$, $G^{k,s}$ is always odd-bipartite. We show that $G^{k,{k \\over 2}}$ is non-odd-bipartite if and only if $G$ is non-bipartite, and find that $G^{k,{k \\over 2}}$ has the same adjacency (respectively, signless Laplacian) spectral radius as $G$. So the results involving the adjacency or signless Laplacian spectral radius of a simple graph $G$ hold for $G^{k,{k \\over 2}}$. In particular, we characterize the unique graph with minimum adjacency or signless Laplacian spectral radius among all non-odd-bipartite hypergraphs $G^{k,{k \\over 2}}$ of fixed order, and prove that $\\sqrt{2+\\sqrt{5}}$ is the smallest limit point of the non-odd-bipartite hypergraphs $G^{k,{k \\over 2}}$. In addition we obtain some results for the spectral radii of the weakly irreducible nonnegative tensors.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2014-08-14T14:45:36.000Z",
      "abstract": "In order to investigate the non-odd-bipartiteness of even uniform hypergraphs, we introduce a class of $k$-uniform hypergraphs $G$, called $(k,\\frac{k}{2})$-hypergraphs, which satisfy the property: $k$ is even, every edge $e$ of $G$ can be divided into two disjoint $\\frac{k}{2}$-vertex sets say $e_{1}$ and $e_{2}$ and for any $e'$ incident to $e$, $e \\cap e'=e_1$ or $e_2$. Such graph $G$ can be constructed from a simple graph, which is called the underlying graph of $G$.We show that $G$ is non-odd-bipartite if and only if the underlying graph of $G$ is non-bipartite. We obtain some results for the spectral radius of weakly irreducible nonnegative tensors, and use them to discuss the perturbation of the spectral radius of the adjacency tensor or signless Laplacian tensor of a $(k,\\frac{k}{2})$-hypergraph after an edge is subdivided. Finally we show that among all $(k,\\frac{k}{2})$-hypergraphs with $n$ half edges, the minimum spectral radius of the adjacency tensor (resp. signless Laplacian tensor) is achieved uniquely for $C_n$ when $n$ is odd and for $C_{n-1}+e$ when $n$ is even.",
      "comment": null,
      "journal": null,
      "doi": null
    },
    {
      "version": "v2",
      "updated": "2015-02-15T02:31:57.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "05C65",
      "15A18",
      "15A69"
    ],
    "keywords": [
      "uniform hypergraphs",
      "signless laplacian tensor",
      "non-odd-bipartite",
      "adjacency tensor",
      "underlying graph"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2014arXiv1408.3303K"
    }
  }
}