Descriptive complexity of controllable graphs

Let G be a graph on n vertices with adjacency matrix A, and let 1 be the all -ones vector. We call G controllable if the set of vectors 1, A1, . . . ,A(n-1)1 spans the whole space R-n. We characterize the isomorphism problem of controllable graphs in terms of other combinatorial, geometric and logical problems. We also describe a polynomial time algorithm for graph isomorphism that works for almost all graphs. (C) 2023 The Authors. Published by Elsevier B.V. This is an open access article under the CC BY-NC-ND license (https://creativecommons.org/licenses/by-nc-nd/4.0).