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

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