Two graphs are isomorphic when one can be transformed into the other by renaming its vertices. Polynomial time algorithms exist for determining isomorphism, but only for some graph types. In general, however, it is not known whether the graph isomorphism problem is P- or NP-complete. The author of this paper introduces graph recurrences, which are sequences of vectors based on the adjacency matrix of a graph and an initial vector. These sequences are then used to introduce and investigate three isomorphic related concepts.

The paper begins by defining how a graph can be determined by a sequence of vectors through graph recurrence. A graph recurrence sequence is based on an initial vector, and the graphs adjacency matrix. The resulting sequence of vectors can then be used to determine whether two graphs are isomorphic. The second concept introduced is how to determine when two graphs are -equivalent. The final concept defines how the vertices of a graph can be separated by a set of vectors. In each section, except the introduction, the author poses open questions that he feels may lead to interesting future research.

Its new viewpoint on how to determine isomorphism is the primary contribution of this paper. Graph recurrence provides a new research vehicle for exploring these types of problems. The author provides many theorems to strengthen the foundations of his concepts, and numerous algorithms for testing graph equivalence and isomorphism. Overall, the author does a great job of introducing and defining graph recurrence, and its value in investigating isomorphism. Based on the number of questions posed at the end of each section, this paper primarily serves to introduce this new research direction, and to encourage additional work in the area.