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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.) |

### Citations:

Zbl 0760.15010; Zbl 0744.15010; Zbl 0731.05023; Zbl 0718.15004; Zbl 0699.15005; Zbl 0696.05049; Zbl 0635.15003
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\textit{L. B. Beasley} and \textit{N. J. Pullman}, Discrete Math. 104, No. 2, 143--157 (1992; Zbl 0776.05068)

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### References:

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