# barcode in ssrs 2008 Figure 15.13 A digraph in Java Encoding EAN / UCC - 13 in Java Figure 15.13 A digraph

Figure 15.13 A digraph
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Theorem 15.6 If G is a digraph with m edges, then the sum of all outdegrees equals m and the sum of all indegrees equals m. Each edge contributes 1 to the total of all outdegrees and 1 to the total of all indegrees. So each total must be m. The complete digraph a the digraph that has a (directed) edge from every vertex to every other vertex. EXAMPLE 15.14 The Complete Digraph on Six Vertices
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The graph shown in Figure 15.14 is the complete digraph on six vertices. It has 15 double-directed edges, so the total number of (one-way) edges is 30, which is n(n 1) = 6(6 1) = 6(5) = 30.
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Theorem 15.7 The number of edges in the complete digraph on n vertices is n(n 1). By Theorem 15.2 on page 286, there are n(n 1)/2 undirected edges on the corresponding complete undirected graph. That makes n(n 1)/2 double-directed edges, so the total number of (one-way) directed edges must be twice that number.
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Figure 15.14 A complete digraph
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Corollary 15.2 The number of edges in any digraph on n vertices is m
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n(n 1).
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Every digraph has an embedded graph, obtained by converting each directed edge into an undirected edge and then removing duplicate edges and loops. Mathematically, this amounts to converting each ordered pair (x, y) of vertices in E into the set {x, y} and then removing all sets of size one (i.e., singletons). EXAMPLE 15.15 The Embedded Graph of a Digraph
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The embedded graph of the digraph in Figure 15.13 is the graph shown in Figure 15.15.
An adjacency matrix for a digraph (V, E) is a two-dimensional boolean array
boolean[][] a;
Figure 15.15 An embedded graph
obtained by ordering the vertices V = {v0 , v1, . . . , vn 1} and then assigning true to a[i][j] if and only if there exists an edge emanating from vertex vi and terminating at vertex vj. EXAMPLE 15.16 An Adjacency Matrix for a Digraph
Figure 15.16 shows the adjacency matrix for the graph in Figure 15.13 on page 292.
Note that the number of true entries in an adjacency matrix for a digraph is equal to the number of edges. Also, as with undirected graphs, different orderings of the vertex set V will result in different adjacency matrices for the same digraph.
int[][] a;