2017/09/25 by Franklin Kenter, Kenter, Franklin H. J., Jephian C.-H. Lin +1 · 2 citations
Computer Science · #Blind Source Separation Techniques #Combinatorics (math.CO) #Discrete Mathematics (cs.DM) #FOS: Computer and information sciences #FOS: Mathematics #Optimization and Control (math.OC)
paper · pdf · doi:10.48550/arxiv.1709.08740
openalex publication_date 2017/09/25 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Zero forcing is an iterative process on a graph used to bound the maximum\nnullity. The process begins with select vertices as colored, and the remaining\nvertices can become colored under a specific color change rule. The goal is to\nfind a minimum set of vertices such that after iteratively applying the rule,\nall of the vertices become colored (i.e., a minimum zero forcing set). Of\nparticular interest is the propagation time of a chosen set which is the number\nof steps the rule must be applied in order to color all the vertices of a\ngraph.\n We give a purely linear algebraic interpretation of zero forcing: Find a set\nof vertices S such that for any weighted adjacency matrix \A,\nwhenever \Ax = \0, the entirety of of \x can be\nrecovered using only \xS, the entries corresponding to S. The key\nhere is that S must be chosen before \A. In this light, we are able\nto give a linear algebraic interpretation of the propagation time: Any error in\n\xS effects the error of \x exponentially in the\npropagation time. This error can be quantitatively measured using newly defined\nzero forcing-related parameters, the error polynomial vector and the variance\npolynomial vector. In this sense, the quality of two zero forcing sets can\nobjectively be compared even if the sets are the same size and their\npropagation time is the same. Examples and constructions are given.\n