{
  "id": "2411.01551",
  "version": "v1",
  "published": "2024-11-03T12:54:43.000Z",
  "updated": "2024-11-03T12:54:43.000Z",
  "title": "New arithmetic invariants for cospectral graphs",
  "authors": [
    "Yizhe Ji",
    "Quanyu Tang",
    "Wei Wang",
    "Hao Zhang"
  ],
  "comment": "13 pages, 1 figure",
  "categories": [
    "math.CO"
  ],
  "abstract": "An invariant for cospectral graphs is a property shared by all cospectral graphs. In this paper, we present three new invariants for cospectral graphs, characterized by their arithmetic nature and apparent novelty. Specifically, let $G$ and $H$ be two graphs with adjacency matrices $A(G)$ and $A(H)$, respectively. We show, among other results, that if $G$ and $H$ are cospectral, then $e^{\\rm T}A(G)^me\\equiv e^{\\rm T}A(H)^m e~({\\rm mod}~4)$ for any integer $m\\geq 0$, where $e$ is the all-one vector. As a simple consequence, we demonstrate that under certain conditions, every graph cospectral with $G$ is determined by its generalized spectrum.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2024-11-03T12:54:43.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "05C50"
    ],
    "keywords": [
      "cospectral graphs",
      "arithmetic invariants",
      "graph cospectral",
      "arithmetic nature",
      "all-one vector"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 13,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}