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

By Wai-Kai Chen

ISBN-10: 0444101055

ISBN-13: 9780444101051

ISBN-10: 0720423627

ISBN-13: 9780720423624

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Content point » Research

Keywords » software of Graph idea for effective research - Finite point research - Meta-heuristic Algorithms

Related matters » Computational Intelligence and Complexity - Computational technology & Engineering

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For our purpose, we shall reserve the symbol Β to represent a submatrix of Ba, of order m χ b and of rank m. The matrix Β is termed as a basis circuit matrix of G because it contains all the information that is contained in Ba. Obviously, the /-circuit matrix is a basis circuit matrix of G. 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.

Show that G is a block if, and only if, for every three distinct nodes of G, there is a path connecting any two of them which contains the third. 22. 21 still true if the last clause is replaced by "which does not contain the t h i r d " ? Can you formulate other equivalent statements? If possible, give four more. 23. Prove that in a nontrivial connected graph having no self-loops there are at least two nodes which are not cutpoint. Is this true for any connected graph? 24. Prove that a nontrivial nonseparable graph contains at least one circuit.

19. A sectional subgraph of the directed graph o f fig. 18. The terms subgraph, sectional subgraph, rank, nullity, the complement of a subgraph, and other graph operations outlined in § 3 can similarly be defined for Gd. The only difference is that we use Gd instead of G. F o r example, in fig. 18 the sectional subgraph Gd[FJ defined by the node set Vs= {1, 2, 3, 6, 7} is presented in fig. 19. 25: Associated undirected graph. To every directed graph Gd, there is an associated undirected graph Gu whose node and edge sets are the same as those in Gd except that the directions of the edges (orders in the pairs of Gd are removed.

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Applied graph theory by Wai-Kai Chen

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