{
  "abstract": "Aldous and Fill (2002) conjectured the asymptotic maximum relaxation time of a random walk on a connected regular graph. Since the relaxation time of a $d$-regular graph $G$ is $d/μ(G)$, where $μ(G)$ denotes its algebraic connectivity, this conjecture is closely related to the problem of minimizing algebraic connectivity among regular graphs. Guiduli and Mohar (1996) conjectured that, for every fixed minimum degree $δ=d\\ge3$ and all sufficiently large orders, graphs with minimum algebraic connectivity are path-like and, apart from bounded portions near their two ends, have a prescribed block structure. Abdi and Ghorbani (2024) proposed an analogous structural conjecture for $d$-regular graphs with minimum algebraic connectivity and fixed degree $d\\ge3$. In Part~I, we proved the Aldous--Fill conjecture, the Guiduli--Mohar conjecture, and for odd degrees, the Abdi Ghorbani conjecture. In this paper, we settle the remaining even-degree case, thereby completing the structural characterization of regular graphs with minimum algebraic connectivity. We also prove that, for every fixed even $d\\ge4$, the minimum algebraic connectivity at order $n$ is $2(d-2)π^2/n^2+O_d(n^{-3})$, and that every minimizing graph has diameter $3n/(d+1)+O_d(1)$. For every fixed even $d\\ge6$, $d$-regular graphs whose algebraic connectivity is asymptotically minimum have asymptotically maximum diameter. Finally, we obtain a sharp normalized-gap bound for all even regular degrees, including degrees that grow with $n$.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.26700",
  "author_total": 2,
  "authors": [
    {
      "name": "Maryam Abdi"
    },
    {
      "name": "Ebrahim Ghorbani"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/6cb9n403j5jk5.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.26700",
  "fulltext_url": null,
  "id": "6cb9n403j5jk5",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.26700",
  "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": "W7214121634",
  "pdf_url": "https://arxiv.org/pdf/2609.26700",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-22",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/6cb9n403j5jk5.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:23:43Z",
      "native_id": "2609.26700",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.26700"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "Graphs with Minimum Algebraic Connectivity II: Regular Graphs of Even Degree",
  "title_text": "Graphs with Minimum Algebraic Connectivity II: Regular Graphs of Even Degree",
  "type": "preprint",
  "updated_at": "2026-10-30T12:23:43Z",
  "url": "https://research.draconitic.com/paper/6cb9n403j5jk5",
  "venue": "arXiv",
  "versions": []
}