## Systems Analysis by Graphs and Matroids: Structural Solvability and ControllabilityRecent technology involves large-scale physical or engineering systems consisting of thousands of interconnected elementary units. This monograph illustrates how engineering problems can be solved using the recent results of combinatorial mathematics through appropriate mathematical modeling. The structural solvability of a system of linear or nonlinear equations as well as the structural controllability of a linear time-invariant dynamical system are treated by means of graphs and matroids. Special emphasis is laid on the importance of relevant physical observations to successful mathematical modelings. The reader will become acquainted with the concepts of matroid theory and its corresponding matroid theoretical approach. This book is of interest to graduate students and researchers. |

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### Contents

Introduction | 1 |

GraphTheoretic Approach to the Solvability | 28 |

GraphTheoretic Approach to the Controllability | 81 |

Copyright | |

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### Common terms and phrases

algebraically independent algorithm arc-set arcs associated bipartite graph assume assumption auxiliary graph auxiliary network block-triangular blocks coefficient matrix column-set columns combinatorial canonical form computation consider corresponding decomposition defined denoted descriptor system 12.3 dimensional analysis Dulmage-Mendelsohn decomposition dynamical system electrical network equivalent Example exists expressed extended form 12.2 feedback vertex set flow f given graph G graph-theoretic independent over Q independent-flow problem Jacobian matrix layered mixed matrix linear M-components mathematical matrix D(s matroid matroid-theoretic max-flow min-cut theorem maximum flow maximum independent flow maximum matching Menger-type complete linking minimum minimum cuts Murota nonsingular nonvanishing entries obtained partial order permutation permutation matrix polymatroid polynomial Proof Prop Proposition rational numbers representation graph resp respect row-set signal-flow graph standard form 12.1 structural analysis structural controllability structural solvability subgraph submatrix subproblems subset system of equations term-rank Theorem transformation vertex vertex-induced subgraph vertex-set vertices