{
  "abstract": "Let $G$ be a graph of order $n$ with adjacency matrix $A_G$, let $\\mathbf e$ denote the all-one vector, and let$W_G=[\\mathbf e,A_G\\mathbf e,\\ldots,A_G^{n-1}\\mathbf e]$ be its walk matrix. We consider the case $\\operatorname{rank}W_G=n-2$, the first corank for which distinct graphs can have the same walk matrix. We give a complete structural description of such pairs. More precisely, if $G$ and $H$ are distinct graphs on the same labelled vertex set and $\\operatorname{rank}W_G=n-2$, then $W_G=W_H$ if and only if $H$ is obtained from $G$ by a reciprocal Wang--Qiu--Hu (WQH) switching. In this case, $A_G-A_H=uv^T+vu^T$, where $u,v\\in\\{0,\\pm1\\}^n$ have disjoint supports and form a basis of $\\ker W_G^T$. We also determine the minimum order at which a non-isomorphic corank-two walk mate can occur: no such pair exists for $n\\le9$, so the previously known $10$-vertex example is sharp. Starting from a labelled realization of that pair, we use singleton union and join operations, together with the graph coronal, to construct connected non-isomorphic pairs with equal corank-two walk matrices for every $n\\ge10$.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.36895",
  "author_total": 2,
  "authors": [
    {
      "name": "Chaochao Zhu"
    },
    {
      "name": "Qin Yue"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/f4w8mz10mmwh5.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.36895",
  "fulltext_url": null,
  "id": "f4w8mz10mmwh5",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.36895",
  "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": "W7214996251",
  "pdf_url": "https://arxiv.org/pdf/2609.36895",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-06",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/f4w8mz10mmwh5.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:27:26Z",
      "native_id": "2609.36895",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.36895"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "A Characterization of Walk-Matrix Equivalence at Corank Two via Reciprocal WQH Switching",
  "title_text": "A Characterization of Walk-Matrix Equivalence at Corank Two via Reciprocal WQH Switching",
  "type": "preprint",
  "updated_at": "2026-10-30T12:27:26Z",
  "url": "https://research.draconitic.com/paper/f4w8mz10mmwh5",
  "venue": "arXiv",
  "versions": []
}