L.A. Bunimovich and B.Z. Webb developed a theory for isospectral graph reduc- tion. This procedure maintains the spectrum of the network’s adjacency matrix up to a set of eigenvalues known beforehand from its graph structure. We make a simple observa- tion regarding the relation between eigenvectors of the original graph and its reduction, that sheds new light on this theory. As an application we propose an updating algorithm for the maximal eigenvector of the Markov matrix associated to a large sparse dynamical network.