{
  "abstract": "Let $G$ be a graph of order $n$, and let $s^+(G)$ and $s^-(G)$ denote the sums of the squares of the positive and negative adjacency eigenvalues of $G$, respectively. Recently, Liu, Tang, and Zhang proved the conjecture of Elphick, Farber, Goldberg, and Wocjan that every connected graph $G$ of order $n$ satisfies $ \\min\\{s^+(G), s^-(G)\\} \\ge n-1. $ For positive square energy, we strengthen this result by showing that every $2$-connected graph $G$ of order $n$ which is not a cycle satisfies $s^+(G)\\ge n$. The formally analogous assertion for $s^-$ is false: the complete graph $K_n$ satisfies $s^-(K_n)=n-1$. We prove a natural counterpart in the triangle-free class: every triangle-free $2$-connected noncycle $G$ satisfies $ \\min\\{s^+(G),s^-(G)\\}>n. $ More generally, it is enough that some maximum-degree vertex of $G$ belongs to no triangle. Together with the exact square energies of cycles, this characterizes the triangle-free $2$-connected graphs for which $s^-(G)\\ge n$; the only exceptions are the cycles $C_{4k+3}$ with $k\\geq1$.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.04069",
  "author_total": 3,
  "authors": [
    {
      "name": "S. Akbari"
    },
    {
      "name": "Fu-Tao Hu"
    },
    {
      "name": "Ya-Yang Liu"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/ffvdy96sxtvgq.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.04069",
  "fulltext_url": null,
  "id": "ffvdy96sxtvgq",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.04069",
  "language": null,
  "licence": "CC0-1.0",
  "licence_label": "CC0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/publicdomain/zero/1.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.CO"
    ],
    [
      "openalex",
      "2608|Graph theory and applications"
    ]
  ],
  "openalex": "W7208708334",
  "pdf_url": "https://arxiv.org/pdf/2609.04069",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-03",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/ffvdy96sxtvgq.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:18:36Z",
      "native_id": "2609.04069",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.04069"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "Positive and Negative Square Energies of $2$-Connected Graphs",
  "title_text": "Positive and Negative Square Energies of $2$-Connected Graphs",
  "type": "preprint",
  "updated_at": "2026-10-30T12:18:36Z",
  "url": "https://research.draconitic.com/paper/ffvdy96sxtvgq",
  "venue": "arXiv",
  "versions": []
}