{
  "abstract": "Let \\( G \\) be a finite group, and consider its common non-zero graph, which we denote by \\( Γ_{nv}(G) \\). In this graph, the vertices represent the non-linear irreducible characters of \\( G \\). There is an edge connecting two distinct vertices \\( χ_1 \\) and \\( χ_2 \\) if there is a vanishing element \\( g \\in G \\) such that \\( χ_1(g) χ_2(g) \\neq 0 \\). A finite group \\( G \\) is described as a strongly \\( \\mathcal{H}_1' \\)-group if the set of vanishing elements of each of its non-linear irreducible character is equal to the set of vanishing elements of \\( G \\). In this paper, we prove Conjecture 1 from \\cite{ourself}, which states that if \\( Γ_{nv}(G) \\) is null (has no edge), then \\( G \\) must either be a strongly \\( \\mathcal{H}_1' \\)-group or a Frobenius group, whose Frobenius complement is isomorphic to \\( Q_8 \\). Additionally, we show that if there are no triangles in \\( Γ_{nv}(G) \\), this implies that the group is solvable. We also look at the situation where the common non-zero graph of a finite group \\( G \\) contains only one edge. This leads us to conclude that \\( G \\) is either isomorphic to \\( S_4 \\); or that the single edge in \\( Γ_{nv}(G) \\) is of the form \\( \\{θ, ζθ\\} \\) for some \\( ζ\\in \\Irr(G/G') \\) and \\( θ\\in \\Irr(G|G') \\), $G$ is a $p$-nilpotent group for some prime $p$ and for all characters \\( χ\\in \\Irr(G|G') - \\{θ, ζθ\\} \\), we find \\( \\Van(χ) = \\Van(G) \\). Lastly, our study also explores groups whose common non-zero graph forms a star shape, concluding that such configurations can only occur if \\( G \\) has exactly one non-linear irreducible character.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.39290",
  "author_total": 2,
  "authors": [
    {
      "name": "Zeinab Akhlaghi"
    },
    {
      "name": "Sesuai Yash Madanha"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/2m678w9jnmmaj.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.39290",
  "fulltext_url": null,
  "id": "2m678w9jnmmaj",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.39290",
  "language": null,
  "licence": "LicenseRef-arXiv-nonexclusive-1.0",
  "licence_label": "arXiv licence",
  "licence_source": "article",
  "licence_url": "http://arxiv.org/licenses/nonexclusive-distrib/1.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.GR"
    ],
    [
      "openalex",
      "2607|Finite Group Theory Research"
    ]
  ],
  "openalex": "W7215052918",
  "pdf_url": "https://arxiv.org/pdf/2609.39290",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-26",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/2m678w9jnmmaj.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:35:40Z",
      "native_id": "2609.39290",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.39290"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.GR",
      "subfield": "algebra-number-theory"
    }
  ],
  "title": "Common non-zero graphs of groups with few edges",
  "title_text": "Common non-zero graphs of groups with few edges",
  "type": "preprint",
  "updated_at": "2026-10-30T12:35:40Z",
  "url": "https://research.draconitic.com/paper/2m678w9jnmmaj",
  "venue": "arXiv",
  "versions": []
}