{
  "abstract": "Let $A$ be the adjacency matrix of a finite graph $Γ$. We say that vertex $u$ of $Γ$ is sedentary if for some constant $C>0$, $|\\exp(itA)_{u,u}|\\geq C$ for all $t\\in\\mathbb{R}$. In this paper, we investigate vertex sedentariness in walk-regular graphs with a focus on Cayley graphs and, more generally, vertex-transitive graphs. In particular, we define the necklace polynomial of an integral graph and use this notion to characterize sedentariness in integral walk-regular graphs. As an application, we use basic tools from algebraic number theory to show that an integral Cayley graph on a group of odd order is sedentary and provide a lower bound on its sedentariness. We also give sufficient conditions such that a unitary Cayley graph over a finite ring is sedentary. Furthermore, we characterize the existence of twin vertices in Cayley graphs and vertex-transitive graphs, and completely determine the occurrence of sedentariness, perfect state transfer and pretty good state transfer in vertex-transitive graphs with twins. Finally, we prove that if $Γ(G,S)$ is a nonempty, integral, normal Cayley graph, then for any prime $p$ with $\\gcd(p, |G|)=1$, there is at least one eigenvalue of $Γ(G,S)$ which is not a multiple of $p$. This generalizes a result of Arnadottir and Godsil, which states that an integral normal Cayley graph of odd order possesses an odd eigenvalue.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2611.00567",
  "author_total": 3,
  "authors": [
    {
      "name": "Karen Meagher"
    },
    {
      "name": "Hermie Monterde"
    },
    {
      "name": "Tung T. Nguyen"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/c9gw1wbprefjs.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2611.00567",
  "fulltext_url": null,
  "id": "c9gw1wbprefjs",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2611.00567",
  "language": null,
  "licence": "CC-BY-4.0",
  "licence_label": "CC BY 4.0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/licenses/by/4.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.CO"
    ]
  ],
  "openalex": null,
  "pdf_url": "https://arxiv.org/pdf/2611.00567",
  "pmcid": null,
  "pmid": null,
  "published": "2026-11-01",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/c9gw1wbprefjs.ris",
  "sources": [
    {
      "first_seen": "2026-11-03T12:21:10Z",
      "native_id": "2611.00567",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2611.00567"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "Sedentariness in Cayley graphs",
  "title_text": "Sedentariness in Cayley graphs",
  "type": "preprint",
  "updated_at": "2026-11-05T15:20:22Z",
  "url": "https://research.draconitic.com/paper/c9gw1wbprefjs",
  "venue": "arXiv",
  "versions": []
}