{
  "id": "2101.03752",
  "version": "v1",
  "published": "2021-01-11T08:18:26.000Z",
  "updated": "2021-01-11T08:18:26.000Z",
  "title": "On the multiplicity of $Aα$-eigenvalues and the rank of complex unit gain graphs",
  "authors": [
    "Aniruddha Samanta",
    "M. Rajesh Kannan"
  ],
  "categories": [
    "math.CO"
  ],
  "abstract": "Let $ \\Phi=(G, \\varphi) $ be a connected complex unit gain graph ($ \\mathbb{T} $-gain graph) on a simple graph $ G $ with $ n $ vertices and maximum vertex degree $ \\Delta $. The associated adjacency matrix and degree matrix are denoted by $ A(\\Phi) $ and $ D(\\Phi) $, respectively. Let $ m_{\\alpha}(\\Phi,\\lambda) $ be the multiplicity of $ \\lambda $ as an eigenvalue of $ A_{\\alpha}(\\Phi) :=\\alpha D(\\Phi)+(1-\\alpha)A(\\Phi)$, for $ \\alpha\\in[0,1) $. In this article, we establish that $ m_{\\alpha}(\\Phi, \\lambda)\\leq \\frac{(\\Delta-2)n+2}{\\Delta-1}$, and characterize the classes of graphs for which the equality hold. Furthermore, we establish a couple of bounds for the rank of $A(\\Phi)$ in terms of the maximum vertex degree and the number of vertices. One of the main results extends a result known for unweighted graphs and simplifies the proof in [15], and other results provide better bounds for $r(\\Phi)$ than the bounds known in [8].",
  "revisions": [
    {
      "version": "v1",
      "updated": "2021-01-11T08:18:26.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "05C50",
      "05C22",
      "05C35"
    ],
    "keywords": [
      "maximum vertex degree",
      "multiplicity",
      "eigenvalue",
      "connected complex unit gain graph",
      "main results extends"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}