Matrix Tree Theorem

The Matrix channelise Theorem Janneke van den Boomen June 29, 2007 The Matrix luff Theorem Janneke van den Boomen Bachelor Thesis Supervisor: Dr. W. Bosma reserve Reader: Dr. A.R.P. van den Essen Opleiding Wiskunde Radboud Universiteit Nijmegen Contents 1 gateway 2 Properties 2.1 Rank . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Determinants . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3 Matrices and trees . . . . . . . . . . . . . . . . . . . . . . . . 2.4 Binet-Cauchy . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 Proof of the Matrix point Theorem 4 Implementation in magma 5 Special formulas 5 7 7 7 8 9 11 12 13 5.1 put down interpret . . . . . . . . . . . . . . . . . . . . . . . . . . 13 5.2 Complete bipartite graph . . . . . . . . . . . . . . . . . . . . . 15 5.3 Wheels . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 6 References 17 4 1 Introduction In a connected graph G, it is (usu whollyy) easy to ?nd a tree that contains each(prenominal) the vertices and some edges of G; such a subgraph is called a a spanning tree. And maybe one nooky ?nd two, or lead such trees. But how many spanning trees does that graph contain? That is what Gustav Robert Kirchho? (1824-1887) was wondering.

Kirchho? was a German physicist, who contributed to the fundamental understanding of electrical circuits, spectroscopy and radiation. Kirchho? introduce an answer to this question, which is formulated in the Matrix shoetree Theorem. By means of this theorem, solutions to (among others) linear resistive electr ical network problems support be expressed ! much easier. To formulate the Matrix Tree Theorem, we ?rst have to de?ne a hyaloplasm AG . De?nition 1.1 let G be a connected graph with n vertices and m edges (numbered arbitrarily). We orient each edge random. The incidence intercellular substance of G is the n × m matrix AG = [aij ] with ? ? +1 if the j th edge is oriented to the ith summit ?1 if the j th edge is oriented away from the ith peak aij =...If you want to get a full essay, order it on our website: OrderCustomPaper.com

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