Download e-book for iPad: Applied graph theory by Wai-Kai Chen

By Wai-Kai Chen

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7 o1 0 0 . 9 we have Γ Α~ Α 1 0 0 1 - --1 - 1 - 1 1 0 0 1 -1 0 0 0 1 — 0 0 — 1 1 0 1 1 0 οι 1 - 1 ο 0 0. ο ο - 1 1 ο ο ο— 1 ο ο — 1 1_ 1. 26) and - 1 - 1 - 1 ο - 1 - 1 ο ο ο 1 1 ο 1 0 0 0 0 1 0 0 0 0 1 0 0 0 0 1. 19). F r o m the example we have just worked out, one soon realizes that the inversion of a major submatrix A12 of A is not a trivial matter. In order to overcome this difficulty, BRANIN [1962] presented an interesting interpretation of the inverse of A12. 18: Node-to-datum path matrix.

Let A12 be the submatrix of A corresponding to the tree t. Also let Ρ 56 ch. 2 F o u n d a t i o n s of electrical network theory be the node-to-datum path matrix of t with reference node n. 30) PTJ<*JK = Κ for /, k= 1, 2 , 1 , where P= and A12 = and ôik is the Kronecker delta. 31) oik. 7, the branch ek partitions the nodes of t into two sets Vl and V2. If i¥=k, then / ? i x and piy are both nonzero or both zero since the two endpoints of the branch ei are both in Vl or in V2. 7 pix and piy must have the same sign, and since axk and ayk are of opposite signs, it follows that 0 Pix<*xh + PiyCtyk 2 3 = ( · 2 ) for ZVÄ:.

So the lemma is proved. 10: The inverse of a major submatrix of the basis incidence matrix A of a connected directed graph with reference node η is equal to the node-todatum path matrix of the tree corresponding to the columns of the major submatrix with η used as the reference node of the tree. Proof. Let A12 be the submatrix of A corresponding to the tree t. Also let Ρ 56 ch. 2 F o u n d a t i o n s of electrical network theory be the node-to-datum path matrix of t with reference node n. 30) PTJ<*JK = Κ for /, k= 1, 2 , 1 , where P= and A12 = and ôik is the Kronecker delta.

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