Cospectrality preserving graph modifications and eigenvector properties via walk equivalence of vertices

Originating from spectral graph theory, cospectrality is a powerful generalization of exchange symmetry and can be applied to all real-valued symmetric matrices. Two vertices an undirected with real edge weights are cospectral iff the underlying weighted adjacency matrix $M$ fulfills $[M^k]_{u,u} = [M^k]_{v,v}$ for non-negative integer $k$, as result any eigenvector $\phi$ has (or, in presence degeneracies, chosen have) definite parity on $u$ $v$. We here show that powers induce further local relations its eigenvectors, also used design preserving modifications. To this end, we introduce concept \emph{walk equivalence} respect multiplets} which special vertex subsets graph. Walk multiplets allow systematic flexible modifications given pair while cospectrality. The set includes addition removal both edges, such topology altered. In particular, prove new connected walk multiplet by suitable connection becomes so-called unrestricted substitution point (USP), meaning arbitrary may it without breaking Also, interconnections between within shown preserve associated Importantly, demonstrate equivalence $u,v$ imposes structure every obeying $\phi_{u} \pm \phi_{v} \ne 0$ (in case specific choice basis needed). Our work paves way flexibly exploiting hidden structural symmetries generic complex network-like systems.