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Connections between vital linkages and zero forcing are established. Specifically, the notion of a rigid linkage is introduced as a special kind of unique linkage and it is shown that spanning forcing paths of a zero forcing process form a spanning rigid linkage and thus a vital linkage. A related generalization of zero forcing that produces a rigid linkage via a coloring process is developed. One of the motivations for introducing zero forcing is to provide an upper bound on the maximum multiplicity of an eigenvalue among the real symmetric matrices described by a graph. Rigid linkages and a related notion of rigid shortest linkages are utilized to obtain bounds on the multiplicities of eigenvalues of this family of matrices.
Zero forcing is a combinatorial game played on a graph with a goal of turning all of the vertices of the graph black while having to use as few unforced moves as possible. This leads to a parameter known as the zero forcing number which can be used t
The zero forcing number Z(G), which is the minimum number of vertices in a zero forcing set of a graph G, is used to study the maximum nullity / minimum rank of the family of symmetric matrices described by G. It is shown that for a connected graph o
Power domination in graphs arises from the problem of monitoring an electric power system by placing as few measurement devices in the system as possible. A power dominating set of a graph is a set of vertices that observes every vertex in the graph,
The ever increasing demands for mobile network access have resulted in a significant increase in bandwidth usage. By improving the system spectral efficiency, multi-way relay networks (MWRNs) provide promising approaches to address this challenge. In
Zero forcing is a combinatorial game played on a graph where the goal is to start with all vertices unfilled and to change them to filled at minimal cost. In the original variation of the game there were two options. Namely, to fill any one single ve