By Wai-Kai Chen

ISBN-10: 0720423627

ISBN-13: 9780720423624

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**Additional resources for Applied graph theory**

**Example text**

We extend the directed-graph concept by permitting several distinct edges with same initial and terminal nodes ; they are called the parallel edges of Gd. The parallel edges directed from node i to node j are denoted by the symbols ···> (hj)k> k^2. 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?

26. Show that a directed graph is strongly connected if, and only if, there exists a closed directed-edge sequence which includes all the edges at least once. 27. Let E' and E" be two directed-edge sequences which have at least one node in common. 28. 29. 30. 31. 32. 33. 34. 35. 36. which includes all the edges of E'\JE". IS E'\JE" strongly connected? Prove that the rank and nullity of a graph G are equal to the sums of the ranks and nullities, respectively, of its components. If a graph G is separable, show that its rank and nullity are invariant under the decomposition of G into its components.

Show that the complement of a p a t h in a circuit is a path. 43. For a given graph G, the block graph i ? ( G ) of G is the graph whose nodes correspond to the blocks of G, and two of these nodes are connected by an edge whenever the corresponding blocks contain a c o m m o n cutpoint of G. , every block of Β is complete (HARARY [ 1 9 6 9 ] ) . CHAPTER 2 F O U N D A T I O N S OF ELECTRICAL NETWORK THEORY One of the most important applications of graph theory in physical science is its use in electrical network theory.