Maximum nullity and zero forcing of circulant graphs

• Main Authors: Duong Linh, Kroschel Brenda K., Riddell Michael, Vander Meulen Kevin N., Van Tuyl Adam
• Format: Article
• Language: English
• Published: De Gruyter 2020-12-01
• Series: Special Matrices
• Subjects: zero forcing; minimum rank; maximum nullity; circulant graph; bipartite graph; graph product; 05c50; 05c75; 05c76; 15a03
• Online Access: https://doi.org/10.1515/spma-2020-0106

The zero forcing number of a graph has been applied to communication complexity, electrical power grid monitoring, and some inverse eigenvalue problems. It is well-known that the zero forcing number of a graph provides a lower bound on the minimum rank of a graph. In this paper we bound and characterize the zero forcing number of various circulant graphs, including families of bipartite circulants, as well as all cubic circulants. We extend the definition of the Möbius ladder to a type of torus product to obtain bounds on the minimum rank and the maximum nullity on these products. We obtain equality for torus products by employing orthogonal Hankel matrices. In fact, in every circulant graph for which we have determined these numbers, the maximum nullity equals the zero forcing number. It is an open question whether this holds for all circulant graphs.