{
  "id": "2010.06378",
  "version": "v1",
  "published": "2020-10-12T15:50:42.000Z",
  "updated": "2020-10-12T15:50:42.000Z",
  "title": "On regular graphs equienergetic with their complements",
  "authors": [
    "Ricardo A. Podestá",
    "Denis E. Videla"
  ],
  "comment": "25 pages, 4 tables",
  "categories": [
    "math.CO"
  ],
  "abstract": "We give necessary and sufficient conditions on the parameters of a regular graph $\\Gamma$ such that $E(\\Gamma)=E(\\bar \\Gamma)$. Then we show that, up to complements, the only bipartite regular graphs equienergetic and non-isospectral with their complements are the crown graphs or $C_4$. Next, for the family of strongly regular graphs $\\Gamma$ we characterize all possible parameters $srg(n,k,e,d)$ such that $E(\\Gamma) = E(\\bar \\Gamma)$. Furthermore, using this, we prove that a strongly regular graph is equienergetic to its complement if and only if it is either a conference graph or else it is a pseudo Latin square graph (i.e.\\@ has $OA$ parameters). We also characterize all complementary equienergetic pairs of graphs of type $C(2)$, $C(3)$ and $C(5)$ in Cameron's hierarchy. Finally, we consider unitary Cayley graphs over rings $G_R=X(R,R^*)$. We showed that if $R$ is a finite artinian ring with an even number of local factors, then $G_R$ is equienergetic with its complement if and only if $R=\\mathbb{F}_{q} \\times \\mathbb{F}_{q'}$ is the product of 2 finite fields.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2020-10-12T15:50:42.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "05C50",
      "05C75",
      "05C92",
      "05E30"
    ],
    "keywords": [
      "strongly regular graph",
      "bipartite regular graphs equienergetic",
      "pseudo latin square graph",
      "complementary equienergetic pairs",
      "parameters"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 25,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}