Observability in Connected Strongly Regular Graphs and Distance Regular Graphs

This paper concerns the study of observability in consensus networks modeled with strongly regular graphs or distance regular graphs. We first give a Kalman-like simple algebraic criterion for observability in distance regular graphs. This criterion consists in evaluating the rank of a matrix built with the components of the Bose-Mesner algebra associated with the considered graph. Then, we define some bipartite graphs that capture the observability properties of the graph to be studied. In particular, we show that necessary and sufficient observability conditions are given by the nullity of the so-called local bipartite observability graph (respectively, the local unfolded bipartite observability graph) for strongly regular graphs (respectively, the distance regular graphs). When the nullity cannot be derived directly from the structure of these bipartite graphs, the rank of the associated bi-adjacency matrix enables evaluating observability. Eventually, as a byproduct of the main results, we show that nonobservability can be stated just by comparing the valency of the graph to be studied with a bound computed from the number of vertices of the graph and its diameter. Similarly, nonobservability can also be stated by evaluating the size of the maximum matching in the aforementioned bipartite graphs.