{
  "id": "2105.02677",
  "version": "v1",
  "published": "2021-05-05T16:06:18.000Z",
  "updated": "2021-05-05T16:06:18.000Z",
  "title": "The scattering matrix with respect to an Hermitian matrix of a graph",
  "authors": [
    "Takashi Komatsu",
    "Norio Konno",
    "Iwao Sato"
  ],
  "comment": "21 pages. arXiv admin note: substantial text overlap with arXiv:1211.4719",
  "categories": [
    "math.CO"
  ],
  "abstract": "Recently, Gnutzmann and Smilansky presented a formula for the bond scattering matrix of a graph with respect to a Hermitian matrix. We present another proof for this Gnutzmann and Smilansky's formula by a technique used in the zeta function of a graph. Furthermore, we generalize Gnutzmann and Smilansky's formula to a regular covering of a graph. Finally, we define an $L$-fuction of a graph, and present a determinant expression. As a corollary, we express the generalization of Gnutzmann and Smilansky's formula to a regular covering of a graph by using its $L$-functions.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2021-05-05T16:06:18.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "05C50",
      "15A15"
    ],
    "keywords": [
      "hermitian matrix",
      "smilanskys formula",
      "bond scattering matrix",
      "zeta function",
      "regular covering"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 21,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}