## Inverse Problems for Electrical NetworksThis book is a very timely exposition of part of an important subject which goes under the general name of “inverse problems”. The analogous problem for continuous media has been very much studied, with a great deal of difficult mathematics involved, especially partial differential equations. Some of the researchers working on the inverse conductivity problem for continuous media (the problem of recovering the conductivity inside from measurements on the outside) have taken an interest in the authors' analysis of this similar problem for resistor networks.The authors' treatment of inverse problems for electrical networks is at a fairly elementary level. It is accessible to advanced undergraduates, and mathematics students at the graduate level. The topics are of interest to mathematicians working on inverse problems, and possibly to electrical engineers. A few techniques from other areas of mathematics have been brought together in the treatment. It is this amalgamation of such topics as graph theory, medial graphs and matrix algebra, as well as the analogy to inverse problems for partial differential equations, that makes the book both original and interesting. |

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### Contents

Introduction | 1 |

Circular Planar Graphs | 11 |

Resistor Networks | 27 |

Harmonic Functions | 59 |

Characterization I | 83 |

Adjoining Edges | 99 |

Characterization II | 109 |

Medial Graphs | 121 |

Recovering a Graph | 149 |

Layered Networks | 173 |

181 | |

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### Common terms and phrases

2-coloring 2-connection 7-harmonic function bilinear form black intervals boundary circle boundary current boundary data boundary edge boundary nodes boundary spike boundary values calculated Chapter circular order circular pair circular planar graph circular planar resistor clockwise column conductivity function conductors connection through G critical graph cut-points deleting denoted edge in G edge joining edge pq ELECTRICAL NETWORKS empty triangle endpoints entry equation equivalence class family of arcs family of chords fc-connection formula geodesics graph G harmonic continuation implies indices integer interior nodes inverse problem Kirchhoff matrix Kirchhoff's Law Lemma lens Let F medial graph network F nodes of G non-singular non-zero notation Ohm's Law planar resistor network positive potential principal flow path Proof Q through G radial real number response matrix A7 Schur complement sequences of boundary shown in Figure shows submatrix subset Suppose F switch uncrossing underlying graph vector vertex vertices