{
  "id": "1709.00792",
  "version": "v1",
  "published": "2017-09-04T02:44:38.000Z",
  "updated": "2017-09-04T02:44:38.000Z",
  "title": "Graphs determined by their $A_α$-spectra",
  "authors": [
    "Huiqiu Lin",
    "Xiaogang Liu",
    "Jie Xue"
  ],
  "comment": "17 pages",
  "categories": [
    "math.CO"
  ],
  "abstract": "Let $G$ be a graph with $n$ vertices, and let $A(G)$ and $D(G)$ denote respectively the adjacency matrix and the degree matrix of $G$. Define $$ A_{\\alpha}(G)=\\alpha D(G)+(1-\\alpha)A(G) $$ for any real $\\alpha\\in [0,1]$. The collection of eigenvalues of $A_{\\alpha}(G)$ together with multiplicities are called the \\emph{$A_{\\alpha}$-spectrum} of $G$. A graph $G$ is said to be \\emph{determined by its $A_{\\alpha}$-spectrum} if all graphs having the same $A_{\\alpha}$-spectrum as $G$ are isomorphic to $G$. We first prove that some graphs are determined by its $A_{\\alpha}$-spectrum for $0\\leq\\alpha<1$, including the complete graph $K_m$, the star $K_{1,n-1}$, the path $P_n$, the union of cycles and the complement of the union of cycles, the union of $K_2$ and $K_1$ and the complement of the union of $K_2$ and $K_1$, and the complement of $P_n$. Setting $\\alpha=0$ or $\\frac{1}{2}$, those graphs are determined by $A$- or $Q$-spectra. Secondly, when $G$ is regular, we show that $G$ is determined by its $A_{\\alpha}$-spectrum if and only if the join $G\\vee K_m$ is determined by its $A_{\\alpha}$-spectrum for $\\frac{1}{2}<\\alpha<1$. Furthermore, we also show that the join $K_m\\vee P_n$ is determined by its $A_{\\alpha}$-spectrum for $\\frac{1}{2}<\\alpha<1$. In the end, we pose some related open problems for future study.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2017-09-04T02:44:38.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "complement",
      "degree matrix",
      "complete graph",
      "adjacency matrix",
      "related open problems"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 17,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}