{
  "id": "1110.2444",
  "version": "v1",
  "published": "2011-10-11T17:30:09.000Z",
  "updated": "2011-10-11T17:30:09.000Z",
  "title": "Graphs with Diameter $n-e$ Minimizing the Spectral Radius",
  "authors": [
    "Jingfen Lan",
    "Linyuan Lu",
    "Lingsheng Shi"
  ],
  "categories": [
    "math.CO"
  ],
  "abstract": "The spectral radius $\\rho(G)$ of a graph $G$ is the largest eigenvalue of its adjacency matrix $A(G)$. For a fixed integer $e\\ge 1$, let $G^{min}_{n,n-e}$ be a graph with minimal spectral radius among all connected graphs on $n$ vertices with diameter $n-e$. Let $P_{n_1,n_2,...,n_t,p}^{m_1,m_2,...,m_t}$ be a tree obtained from a path of $p$ vertices ($0 \\sim 1 \\sim 2 \\sim ... \\sim (p-1)$) by linking one pendant path $P_{n_i}$ at $m_i$ for each $i\\in\\{1,2,...,t\\}$. For $e=1,2,3,4,5$, $G^{min}_{n,n-e}$ were determined in the literature. Cioab\\v{a}-van Dam-Koolen-Lee \\cite{CDK} conjectured for fixed $e\\geq 6$, $G^{min}_{n,n-e}$ is in the family ${\\cal P}_{n,e}=\\{P_{2,1,...1,2,n-e+1}^{2,m_2,...,m_{e-4},n-e-2}\\mid 2<m_2<...<m_{e-4}<n-e-2\\}$. For $e=6,7$, they conjectured $G^{min}_{n,n-6}=P^{2,\\lceil\\frac{D-1}{2}\\rceil,D-2}_{2,1,2,n-5}$ and $G^{min}_{n,n-7}=P^{2,\\lfloor\\frac{D+2}{3}\\rfloor,D- \\lfloor\\frac{D+2}{3}\\rfloor, D-2}_{2,1,1,2,n-6}$. In this paper, we settle their three conjectures positively. We also determine $G^{min}_{n,n-8}$ in this paper.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2011-10-11T17:30:09.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "05C50"
    ],
    "keywords": [
      "minimal spectral radius",
      "largest eigenvalue",
      "pendant path",
      "adjacency matrix",
      "minimizing"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2011arXiv1110.2444L"
    }
  }
}