{
  "id": "2005.01037",
  "version": "v1",
  "published": "2020-05-03T10:08:01.000Z",
  "updated": "2020-05-03T10:08:01.000Z",
  "title": "On $α$-adjacency energy of graphs and Zagreb index",
  "authors": [
    "S. Pirzada",
    "Bilal A. Rather",
    "Hilal A. Ganie",
    "Rezwan ul Shaban"
  ],
  "comment": "17 pages",
  "categories": [
    "math.CO"
  ],
  "abstract": "Let $A(G)$ be the adjacency matrix and $D(G)$ be the diagonal matrix of the vertex degrees of a simple connected graph $G$. Nikiforov defined the matrix $A_{\\alpha}(G)$ of the convex combinations of $D(G)$ and $A(G)$ as $A_{\\alpha}(G)=\\alpha D(G)+(1-\\alpha)A(G)$, for $0\\leq \\alpha\\leq 1$. If $ \\rho_{1}\\geq \\rho_{2}\\geq \\dots \\geq \\rho_{n}$ are the eigenvalues of $A_{\\alpha}(G)$ (which we call $\\alpha$-adjacency eigenvalues of $G$), the $ \\alpha $-adjacency energy of $G$ is defined as $E^{A_{\\alpha}}(G)=\\sum_{i=1}^{n}\\left|\\rho_i-\\frac{2\\alpha m}{n}\\right|$, where $n$ is the order and $m$ is the size of $G$. We obtain the upper and lower bounds for $E^{A_{\\alpha}}(G) $ in terms of order $n$, size $m$ and Zagreb index $Zg(G)$ associated to the structure of $G$. Further, we characterize the extremal graphs attaining these bounds.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2020-05-03T10:08:01.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "05C50",
      "05C12",
      "15A18"
    ],
    "keywords": [
      "adjacency energy",
      "zagreb index",
      "extremal graphs",
      "diagonal matrix",
      "vertex degrees"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 17,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}