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Controllability of NEPSes of graphs. (English) Zbl 1492.05088

Summary: If \(G\) is a graph with \(n\) vertices, \(A_G\) is its adjacency matrix and \(\mathfrak{b}\) is a binary vector of length \(n\), then the pair \((A_G, \mathfrak{b})\) is said to be controllable (or \(G\) is said to be controllable for the vector \(\mathfrak{b})\) if \(A_G\) has no eigenvector orthogonal to \(\mathfrak{b}\). In particular, if \(\mathfrak{b}\) is the all-1 vector \(\mathfrak{j}\), then we simply say that \(G\) is controllable. In this paper, we consider the controllability of non-complete extended \(p\)-sums (for short, NEPSes) of graphs. We establish some general results and then focus the attention to the controllability of paths and related NEPSes. Moreover, the controllability of Cartesian products and tensor products is also considered. Certain related results concerning signless Laplacian matrices and signed graphs are reported.

MSC:

05C50 Graphs and linear algebra (matrices, eigenvalues, etc.)
05C76 Graph operations (line graphs, products, etc.)
05C22 Signed and weighted graphs
93B05 Controllability
93C05 Linear systems in control theory
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