DisCoMath Seminar: Minimum Zero-Diagonal Rank and Failed Skew Zero Forcing of Graphs
Minimum Zero-Diagonal Rank and Failed Skew Zero Forcing of Graphs
Dr. Bonnie Jacob
Science & Mathematics Department, NTID
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Associated with any simple graph G is a family of symmetric zero-diagonal matrices with the same zero-nonzero pattern as the adjacency matrix of G. There is a strong connection between the ranks of these matrices and the generalized cycles that exist as subgraphs of G. In this talk, we characterize all connected graphs G with minimum rank of 3 or below, as well as all connected graphs with minimum rank of n, the order of the graph. It turns out that the minimum rank is the order of the graph if and only if G has a unique spanning generalized cycle, also known as a unique perfect [1,2]-factor, among other names. We present an algorithm for determining whether a graph has a unique spanning generalized cycle. We also determine the maximum zero-diagonal rank of a graph, which is also related to generalized cycles, and then show that there exist graphs G for which some ranks between minimum rank and the maximum rank of G cannot be realized. Related to the notion of minimum rank is the idea of zero forcing. For minimum zero-diagonal rank, the skew zero forcing number and failed skew zero forcing number provide bounds on the rank of a graph. We present some results for the failed skew zero forcing number of a graph.
Dr. Jacob is an Associate Professor in the Science and Mathematics Department at the National Technical Institute for the Deaf, Rochester Institute of Technology Her research interests include: Graph Theory, Zero Forcing, Graph Labeling Problems, Minimum Rank Problems, and Agent-based Modeling.
Undergraduates, graduates, and experts. Those with interest in the topic.
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When and Where
This is an RIT Only Event