By Wai-Kai Chen

ISBN-10: 0444101055

ISBN-13: 9780444101051

ISBN-10: 0720423627

ISBN-13: 9780720423624

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3: There is a nonsingular m x m matrix C of Ts, — Γ s and O's such that B = CBf. 5: A square submatrix of a basis circuit matrix Β of a connected directed graph is a major submatrix if, and only if, the columns of this submatrix correspond to the edges of a cotree in the directed graph. Proof. Necessity. By permuting the columns of Β if necessary, we may assume that the first m columns of Β form a major submatrix. 9) where B±1 is nonsingular and of order m. Since there are r columns in B12, it is sufficient to show that the subgraph corresponding to these columns does not contain any circuit.

The necessary part is obvious. To prove sufficiency, let us assume that a graph G(V, E) has this property. Without loss of generality, we may further assume that G is connected; for if not we can consider the components of G separately. Also we can assume that G has no self-loops. Let / be a node of G, and let V2 be the set consisting of all the nodes of G that are of even distance from / (node i is in V2), while V1 = V— V2. Evidently, each node in V± is of odd distance from / in G. To show that every edge (w, v) of G is connected between a node u in V2 and a node υ in Vl9 we first assume that both u and ν are in V2.

Also we can assume that G has no self-loops. Let / be a node of G, and let V2 be the set consisting of all the nodes of G that are of even distance from / (node i is in V2), while V1 = V— V2. Evidently, each node in V± is of odd distance from / in G. To show that every edge (w, v) of G is connected between a node u in V2 and a node υ in Vl9 we first assume that both u and ν are in V2. Let Piu and P f* be the shortest paths of even lengths from i to u and v, respectively. Starting from the nodes u and v, let j be the first common node in Piu and P * .

### Applied graph theory by Wai-Kai Chen

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