{
  "id": "1801.02784",
  "version": "v1",
  "published": "2018-01-09T03:45:22.000Z",
  "updated": "2018-01-09T03:45:22.000Z",
  "title": "Spectral Radius of $\\{0, 1\\}$-Tensor with Prescribed Number of Ones",
  "authors": [
    "Shuliang Bai",
    "Linyuan Lu"
  ],
  "comment": "29 pages",
  "categories": [
    "math.CO"
  ],
  "abstract": "For any $r$-order $\\{0, 1\\}$-tensor $A$ with $e$ ones, we prove that the spectral radius of $A$ is at most $e^{\\frac{r-1}{r}}$ with the equality holds if and only if $e={k^r}$ for some integer $k$ and all ones forms a principal sub-tensor ${\\bf 1}_{k\\times \\cdots \\times k}$. We also prove a stability result for general tensor $A$ with $e$ ones where $e=k^r+l$ with relatively small $l$. Using the stability result, we completely characterized the tensors achieving the maximum spectral radius among all $r$-order $\\{0, 1\\}$-tensor $A$ with $k^r+l$ ones, for $-r-1\\leq l \\leq r$, and $k$ sufficiently large.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2018-01-09T03:45:22.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "05C50",
      "05C35"
    ],
    "keywords": [
      "prescribed number",
      "stability result",
      "maximum spectral radius",
      "equality holds",
      "general tensor"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 29,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}