通过互反WQH切换刻画余秩为二的游走矩阵等价性
A Characterization of Walk-Matrix Equivalence at Corank Two via Reciprocal WQH Switching
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中文总结 AI 辅助
本文刻画了余秩为二的游走矩阵等价性，证明不同图具有相同游走矩阵当且仅当通过互反WQH切换得到，并构造了所有$n\geq 10$阶的连通非同构例子，反驳了Liu和Siemons的猜想。
中文摘要 AI 辅助
设$G$是一个阶为$n$的图，其邻接矩阵为$A_G$，令$\mathbf e$表示全一向量，并令$W_G=[\mathbf e,A_G\mathbf e,\ldots,A_G^{n-1}\mathbf e]$为其游走矩阵。我们考虑$\operatorname{rank}W_G=n-2$的情形，这是第一个使得不同图可以具有相同游走矩阵的余秩。我们给出了此类图对的完整结构描述。更精确地，若$G$和$H$是同一标号顶点集上的不同图，且$\operatorname{rank}W_G=n-2$，则$W_G=W_H$当且仅当$H$可由$G$通过一次互反Wang--Qiu--Hu（WQH）切换得到。在此情形下，$A_G-A_H=uv^T+vu^T$，其中$u,v\in\{0,\pm1\}^n$具有不相交的支撑集，并构成$\ker W_G^T$的一组基。我们还确定了具有相等余秩二游走矩阵的非同构图对可能出现的最小阶数。对于$n\leq 9$，不存在这样的图对，而在$10$个顶点上存在一个连通图对。从这一例子出发，我们利用单点并和联运算，以及图的冠（coronal），为每个$n\geq 10$构造具有相等余秩二游走矩阵的连通非同构图对。这特别地反驳了Liu和Siemons的一个猜想。
英文摘要
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 $n\ge10$.
发表机构
- College of Finance and Mathematics, West Anhui University（皖西学院金融数学学院）
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