{
  "abstract": "We study how irreducible factors associated with main eigenvalues constrain the connected components of generalized cospectral mates. Let $M_G(x)$ be the main polynomial of a graph $G$, and let $κ_{\\mathrm m}(G)$ denote the sum of the multiplicities in $ϕ_G(x)$ of the irreducible factors dividing $M_G(x)$. We prove that every graph $H$ with the same characteristic polynomial and main polynomial as $G$ satisfies $c(H)\\leq κ_{\\mathrm m}(G)$. Consequently, if $T$ is a tree and $H$ is generalized cospectral with $T$, then $β(H)=c(H)-1\\leq κ_{\\mathrm m}(T)-1$. We develop the case $κ_{\\mathrm m}(T)=2$ in detail. Any disconnected generalized cospectral mate is the union of a tree and a connected bipartite unicyclic graph, and the coronals of its two components are uniquely prescribed by the canonical decomposition of the coronal of $T$ with respect to the two irreducible main factors. This turns the existence of a disconnected mate into a component-realizability problem. Factor moments yield realizability and tree-forcing obstructions, while matching data provide complementary characteristic-polynomial information. In particular, the unique cycle of any disconnected mate has length at least $6$, and finitely many matching identities, together with component-coronal realizability, certify generalized cospectrality under an explicit degree bound. As an application, we show that the double star $D(2m,m+1)$ is DGS but not DS whenever $m\\geq2$ and neither $m$ nor $2m+2$ is a perfect square. In particular, $D(8t+4,4t+3)$, $t\\geq0$, gives an explicit infinite family of such graphs.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2608.24137",
  "author_total": 1,
  "authors": [
    {
      "name": "Chaochao Zhu"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/4fddzr6mfx4tc.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2608.24137",
  "fulltext_url": null,
  "id": "4fddzr6mfx4tc",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2608.24137",
  "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.CO"
    ],
    [
      "openalex",
      "2608|Graph theory and applications"
    ]
  ],
  "openalex": "W7204256700",
  "pdf_url": "https://arxiv.org/pdf/2608.24137",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-07",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/4fddzr6mfx4tc.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:19:58Z",
      "native_id": "2608.24137",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2608.24137"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "Main-Factor Allocation and Coronal Realizability for Generalized Cospectral Mates of Trees",
  "title_text": "Main-Factor Allocation and Coronal Realizability for Generalized Cospectral Mates of Trees",
  "type": "preprint",
  "updated_at": "2026-10-30T12:19:58Z",
  "url": "https://research.draconitic.com/paper/4fddzr6mfx4tc",
  "venue": "arXiv",
  "versions": []
}