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By Wai-Kai Chen

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If no particular edge is specified, (/, j) denotes any one, but otherwise fixed, of the parallel edges from / to j in Gd. Also we admit edges with the same endpoints; they are called the self loops of Gd. 25 Directed graphs §5 As an illustration, consider the directed graph Gd(V, E) in which V = { 1 , 2 , 3, 4, 5, 6, 7 } , Ε = {(1, 1), (1, 2), ( 1 , 4), (4, 4 ) 1? ( 6 , 7 ) 2 , (7, 6 ) } . The corresponding geometric graph is as shown in fig. 18 in which we have a self-loop at node 1 and two parallel edges directed from node 2 to node 3, and from node 6 to node 7.

In Maxwell's formulation, however, it is necessary to choose an appropriate set of circuits or node-pairs as independent variables and also to assign an orientation to each branch of the network. It appears, therefore, 36 §1 Matrices and directed graphs 37 that the branch voltages and currents induced in the branches of the network may depend upon the choice of circuits or node-pairs and the assignment of branch orientations. Such a dependence would contradict the known electrical network situation.

5 : A cut and a circuit of G have an even number of edges in common. The above two lemmas follow directly from our interpretation of a cut. 5). Once again, we can discuss cuts most conveniently by means of a cut matrix. 15: Cut-edge incidence matrix (cut matrix). The cut-edge incidence matrix or simply the cut matrix, denoted by the symbol Qa, of a directed graph G is a matrix of order q χ b, where q is the number of nonempty cuts in G, such that if ö a = l > u ] > t h ne if edge ej is in cut / and the orientations of the cut and the edge coincide, qu = — 1 if edge ej is in cut / and the orientations of the cut and the edge are opposite, qij = 0 if the edge ej is not in cut /.

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