A selected tour of the theory of identification matrices

A selected tour of the theory of identification matrices is offered here. We show that, among other things, shortest-path adjacency matrices are identification matrices for all simple graphs and adjacency matrices are identification matrices for all bipartite graphs. Additionally, we provide an improved proof that augmented adjacency matrices satisfying the circular 1's property are identification matrices. We also present a characterization of doubly convex bipartite graphs by identification matrices. Based on the theory of identification matrices, we describe an improved method for testing isomorphism between Γ circular arc graphs. The sequential algorithm can be implemented to run in O(n²) time and is optimal if the graphs are given as (augmented) adjacency matrices, so to speak.