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Linear operators that strongly preserve graphical properties of matrices. (English) Zbl 0776.05068

Summary: An operator on the set \({\mathcal M}\) of \(n\times n\) matrices strongly preseveres a subset \({\mathcal F}\) if it maps \({\mathcal F}\) into \({\mathcal F}\) and \({\mathcal M}\backslash{\mathcal F}\) into \({\mathcal M\backslash F}\). The operator semigroup of \({\mathcal F}\) is the semigroup of linear operators strongly preserving \({\mathcal F}\). We show that all the \(n\times n\) matrix-families which are determined by the directed graphs of their members and satisfy a short list of conditions, have the same operator semigroup \(\Sigma\), and we determine the generators of \(\Sigma\). Among those matrix-families are: the irreducible matrices; the matrices whose directed graphs have maximum cycle length \(l\geq k\) for fixed \(k\geq 4\); and the matrices whose directed graphs have a path of length at least \(l\geq k\) for fixed \(k\geq 3\). Similar results are obtained for matrix-families determined by the undirected graphs of their members.

MSC:

05C50 Graphs and linear algebra (matrices, eigenvalues, etc.)
15A04 Linear transformations, semilinear transformations
05B20 Combinatorial aspects of matrices (incidence, Hadamard, etc.)
05C38 Paths and cycles
15B33 Matrices over special rings (quaternions, finite fields, etc.)
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References:

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