Graph recurrence

For a graph G, a graph recurrence sequence x0,x1,x2,… of vectors is defined by the recurrencext+1=Axt, t=0,1,…,where A is the adjacency matrix of G and x0 is an initial vector. Each vector in this sequence can be thought of as a vertex labeling of G, the label at a given vertex at step t+1 obtained by summing the values at the adjacent vertices at step t. Based on graphical sequences, three concepts are defined: (1) for a graph to be determined by a set of vectors, (2) for two graphs to be m-equivalent, and (3) for the vertices of the graph to be separated by a set of vectors. Results concerning these notions are given, relations to the graph isomorphism problem are discussed, and numerous open problems are posed.