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

Abstract

Let $G$ be a graph of order $n$ with adjacency matrix $A_G$, let $\mathbf{e}$ denote the all-one vector, and let$W_G=[\mathbf{e},A_G\mathbf{e},\dots ,A_G^{n-1}\mathbf{e}]$ be its walk matrix. We consider the case $\mathrm{rank}{W}_G=n-2$, 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 $\mathrm{rank}{W}_G=n-2$, 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, $A_G-A_H=u{v}^T+v{u}^T$, where $u,v\in \{0,\pm 1{\}}^n$ 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 $n\le 9$, 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 $n\ge 10$.

arXiv commentV2: Added the prior 10-vertex counterexample of Lv--Liu--Wang--Wang and clarified the novelty claims. The main structural characterization, minimum-order proof, and infinite-family results are unchanged