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A Characterization of Walk-Matrix Equivalence at Corank Two via Reciprocal WQH Switching

  • Chaochao Zhu
  • Qin Yue
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Abstract

Let G be a graph of order n with adjacency matrix AG, let e denote the all-one vector, and letWG=[e,AGe,…,AGn−1e] be its walk matrix. We consider the case rankWG=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 rankWG=n−2, then WG=WH if and only if H is obtained from G by a reciprocal Wang--Qiu--Hu (WQH) switching. In this case, AG−AH=uvT+vuT, where u,v∈{0,±1}n have disjoint supports and form a basis of ker WGT. We also determine the minimum order at which a non-isomorphic corank-two walk mate can occur: no such pair exists for n≤9, 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≥10.

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Cite

BibTeXRIS
BibTeX
@misc{zhu2026characterization,
  title = {{A Characterization of Walk-Matrix Equivalence at Corank Two via Reciprocal WQH Switching}},
  author = {Chaochao Zhu and Qin Yue},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2609.36895},
  eprint = {2609.36895},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.36895},
}
RIS
TY  - UNPB
TI  - A Characterization of Walk-Matrix Equivalence at Corank Two via Reciprocal WQH Switching
AU  - Chaochao Zhu
AU  - Qin Yue
PY  - 2026
DA  - 2026/10/06
PB  - arXiv
DO  - 10.48550/arxiv.2609.36895
UR  - https://arxiv.org/abs/2609.36895
AB  - Let G be a graph of order n with adjacency matrix A_G, let e denote the all-one vector, and letW_G=[e,A_Ge,…,A_G^n−1e] be its walk matrix. We consider the case rankW_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 rankW_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∈{0,±1}^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≤9, 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≥10.
ER  -