{
  "id": "2409.18220",
  "version": "v1",
  "published": "2024-09-26T18:58:48.000Z",
  "updated": "2024-09-26T18:58:48.000Z",
  "title": "A Linear Lower Bound for the Square Energy of Graphs",
  "authors": [
    "Saieed Akbari",
    "Hitesh Kumar",
    "Bojan Mohar",
    "Shivaramakrishna Pragada"
  ],
  "comment": "5 pages, 1 figure",
  "categories": [
    "math.CO"
  ],
  "abstract": "Let $G$ be a graph of order $n$ with eigenvalues $\\lambda_1 \\geq \\cdots \\geq\\lambda_n$. Let \\[s^+(G)=\\sum_{\\lambda_i>0} \\lambda_i^2, \\qquad s^-(G)=\\sum_{\\lambda_i<0} \\lambda_i^2.\\] The smaller value, $s(G)=\\min\\{s^+(G), s^-(G)\\}$ is called the \\emph{square energy} of $G$. In 2016, Elphick, Farber, Goldberg and Wocjan conjectured that for every connected graph $G$ of order $n$, $s(G)\\geq n-1.$ No linear bound for $s(G)$ in terms of $n$ is known. Let $H_1, \\ldots, H_k$ be disjoint vertex-induced subgraphs of $G$. In this note, we prove that \\[s^+(G)\\geq\\sum_{i=1}^{k} s^+(H_i) \\quad \\text{ and } \\quad s^-(G)\\geq\\sum_{i=1}^{k} s^-(H_i),\\] which implies that $s(G)\\geq \\frac{3n}{4}$ for every connected graph $G$ of order $n\\ge 4$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2024-09-26T18:58:48.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "05C50"
    ],
    "keywords": [
      "linear lower bound",
      "square energy",
      "connected graph",
      "smaller value",
      "linear bound"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 5,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}