{
  "id": "1304.4696",
  "version": "v1",
  "published": "2013-04-17T05:42:28.000Z",
  "updated": "2013-04-17T05:42:28.000Z",
  "title": "Spectral moments of trees with given degree sequence",
  "authors": [
    "Eric Ould Dadah Andriantiana",
    "Stephan Wagner"
  ],
  "comment": "24 pages 5 figures",
  "categories": [
    "math.CO"
  ],
  "abstract": "Let $\\lambda_1,\\dots,\\lambda_n$ be the eigenvalues of a graph $G$. For any $k\\geq 0$, the $k$-th spectral moment of $G$ is defined by $\\M_k(G)=\\lambda_1^k+\\dots+\\lambda_n^k$. We use the fact that $\\M_k(G)$ is also the number of closed walks of length $k$ in $G$ to show that among trees $T$ whose degree sequence is $D$ or majorized by $D$, $\\M_k(T)$ is maximized by the greedy tree with degree sequence $D$ (constructed by assigning the highest degree in $D$ to the root, the second-, third-, \\dots highest degrees to the neighbors of the root, and so on) for any $k\\geq 0$. Several corollaries follow, in particular a conjecture of Ili\\'c and Stevanovi\\'c on trees with given maximum degree, which in turn implies a conjecture of Gutman, Furtula, Markovi\\'c and Gli\\v{s}i\\'c on the Estrada index of such trees, which is defined as $\\EE(G)=e^{\\lambda_1}+\\dots+e^{\\lambda_n}$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2013-04-17T05:42:28.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "68R05",
      "05C05",
      "05C35",
      "05C50",
      "G.2.1",
      "G.2.2"
    ],
    "keywords": [
      "degree sequence",
      "highest degree",
      "th spectral moment",
      "estrada index",
      "turn implies"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 24,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2013arXiv1304.4696O"
    }
  }
}