{
  "id": "1912.10645",
  "version": "v1",
  "published": "2019-12-23T06:46:56.000Z",
  "updated": "2019-12-23T06:46:56.000Z",
  "title": "On the Hopf algebra of graphs",
  "authors": [
    "Miodrag Iovanov",
    "Jaiung Jun"
  ],
  "categories": [
    "math.CO"
  ],
  "abstract": "The algebra of graphs is defined as the algebra which has a formal basis $\\mathcal{G}$ of all isomorphism types of graphs, and multiplication is to take the disjoint union. We explicitly describe here the structure of the Hopf algebra of graphs $H$. We find an explicit basis $\\mathcal{B}$ of the space of primitives, such that each graph is a polynomial with non-negative integer coefficients of the elements of $\\mathcal{B}$, and each $b\\in\\mathcal{B}$ is a polynomial with integer coefficients in $\\mathcal{G}$. Using this, we find the cancellation and grouping free formula for the antipode. The coefficients appearing in all these polynomials are, up to signs, numbers counting multiplicities of subgraphs in a graph. We then investigate applications of this to the graph reconstruction conjectures, and rederive some results in the literature on these questions.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2019-12-23T06:46:56.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "hopf algebra",
      "graph reconstruction conjectures",
      "polynomial",
      "grouping free formula",
      "explicit basis"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}