Applied graph theory by Wai-Kai Chen

By Wai-Kai Chen

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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:. If i = k, two cases are considered: pkx = 0 and / > f c = x ± l .

Since there are r columns in B12, it is sufficient to show that the subgraph corresponding to these columns does not contain any circuit. Assume otherwise, and let Bt be the row in Ba corresponding to this circuit. Consider the submatrix of Ba consisting of the rows of Β a n d Βi. 10) is ( r a + 1 ) . 4 is of rank m. Hence this is impossible. Sufficiency. Assume that the columns of a submatrix of Β correspond to a cotree. Let Bf be the /-circuit matrix formed by the chords of this cotree in the directed graph G.

2 . 3 with the orientations removed is the set of trees of Gu. Obviously, a tree of a connected graph has r( = n— 1 ) edges. 40 F o u n d a t i o n s of electrical network theory ch. 2 The concept of trees is extremely important because of the number of properties of the network that can be related to a tree. For example, the number of independent Kirchhoff 's voltage equations, the number of state equations, the methods of choosing the independent equations, and the topological formulas for network functions, may all be stated in terms of trees.

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