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 $W_G=[\mathbf e,A_G\mathbf e,\ldots,A_G^{n-1}\mathbf e]$ 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 pair with equal corank-two walk matrices can occur. No such pair exists for $n\leq 9$, while a connected pair exists on $10$ vertices. Starting from this example, we use singleton union and join operations, together with the graph coronal, to construct connected non-isomorphic pairs with equal walk matrices of corank two for every $n\geq 10$. This, in particular, disproves a conjecture of Liu and Siemons.