{
  "id": "math/0610742",
  "version": "v3",
  "published": "2006-10-25T04:23:20.000Z",
  "updated": "2007-08-30T03:33:36.000Z",
  "title": "Clustering of spectra and fractals of regular graphs",
  "authors": [
    "V. Ejov",
    "J. A. Filar",
    "S. K. Lucas",
    "P. Zograf"
  ],
  "comment": "10 pages, 5 figures",
  "journal": "J. Math. Anal. Appl. 333 (2007) 236-246",
  "categories": [
    "math.CO",
    "math.ST",
    "stat.TH"
  ],
  "abstract": "We exhibit a characteristic structure of the class of all regular graphs of degree d that stems from the spectra of their adjacency matrices. The structure has a fractal threadlike appearance. Points with coordinates given by the mean and variance of the exponentials of graph eigenvalues cluster around a line segment that we call a filar. Zooming-in reveals that this cluster splits into smaller segments (filars) labeled by the number of triangles in graphs. Further zooming-in shows that the smaller filars split into subfilars labelled by the number of quadrangles in graphs, etc. We call this fractal structure, discovered in a numerical experiment, a multifilar structure. We also provide a mathematical explanation of this phenomenon based on the Ihara-Selberg trace formula, and compute the coordinates and slopes of all filars in terms of Bessel functions of the first kind.",
  "revisions": [
    {
      "version": "v3",
      "updated": "2007-08-30T03:33:36.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "05C50",
      "62P99"
    ],
    "keywords": [
      "regular graphs",
      "ihara-selberg trace formula",
      "graph eigenvalues cluster",
      "smaller filars split",
      "multifilar structure"
    ],
    "tags": [
      "journal article"
    ],
    "publication": {
      "journal": "Journal of Mathematical Analysis and Applications",
      "year": 2007,
      "month": "Sep",
      "volume": 333,
      "number": 1,
      "pages": 236
    },
    "note": {
      "typesetting": "TeX",
      "pages": 10,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2007JMAA..333..236E"
    }
  }
}