© 2014 Elsevier Inc. Centraliser codes are codes of length n2 defined as centralisers of a given matrix A of order n. Their dimension, parity-check matrices, syndromes, and automorphism groups are investigated. A lower bound on the dimension is n, the order of A. This bound is met when the minimal polynomial is equal to the annihilator, i.e. for so-called cyclic (a.k.a. non-derogatory) matrices. If, furthermore, the matrix is separable and the adjacency matrix of a graph, the automorphism group of that graph is shown to be abelian and to be even trivial if the alphabet field is of even characteristic.