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barcode in ssrs 2008 Figure 15.1 A graph in Java
Figure 15.1 A graph EAN13 Decoder In Java Using Barcode Control SDK for Java Control to generate, create, read, scan barcode image in Java applications. Encoding EAN13 In Java Using Barcode generation for Java Control to generate, create UPC  13 image in Java applications. Note that by definition an edge is a set with exactly two elements. This prevents the possibility of a loop being an edge because a loop involves only one vertex. So the definition of simple graphs excludes the possibility of loops. Also note that since E is a set, an edge cannot be listed more than once. (Sets do not allow repeated members.) So the definition of simple graphs excludes the possibility of multiple edges. In general, graphs may include loops and multiple edges; simple graphs do not. GRAPH TERMINOLOGY If G = (V, E) is a graph and G = (V , E ) where V V and E of G. If V = V, then G is called a spanning subgraph of G. Every graph is a spanning subgraph of itself. E, then G is called a subgraph Scan EAN13 In Java Using Barcode reader for Java Control to read, scan read, scan image in Java applications. Barcode Generation In Java Using Barcode creation for Java Control to generate, create barcode image in Java applications. EXAMPLE 15.2 Subgraphs
Bar Code Reader In Java Using Barcode recognizer for Java Control to read, scan read, scan image in Java applications. Encoding EAN13 In Visual C# Using Barcode drawer for .NET Control to generate, create GTIN  13 image in .NET applications. GRAPHS
European Article Number 13 Creator In Visual Studio .NET Using Barcode drawer for ASP.NET Control to generate, create EAN13 image in ASP.NET applications. European Article Number 13 Encoder In .NET Using Barcode creation for VS .NET Control to generate, create EAN / UCC  13 image in Visual Studio .NET applications. [CHAP. 15
Encode EAN13 In Visual Basic .NET Using Barcode generation for VS .NET Control to generate, create EAN13 Supplement 5 image in .NET framework applications. EAN13 Maker In Java Using Barcode encoder for Java Control to generate, create EAN13 image in Java applications. The graph G1 = (V1, E1) in Figure 15.2 with vertex set V1 = {a, b, d} and edge set E1 = {ad, bd} is a nonspanning subgraph of the graph in Example 15.1. This subgraph has size 3. The graph G2 = (V2, E2) in Figure 15.2 with vertex set V2 = {a, b, c, d} and edge set E2 = {ab, ac, cd} is a spanning subgraph of the graph in Example 15.1. This subgraph has size 4. UPCA Supplement 2 Creator In Java Using Barcode drawer for Java Control to generate, create UPC A image in Java applications. Bar Code Generation In Java Using Barcode drawer for Java Control to generate, create bar code image in Java applications. The degree (or valence) of a vertex is the number of edges that are incident upon it. For example, in the graph in Figure 15.1 on page 285, vertex a has degree 3 and vertex b has degree 2. But in the subgraph G1 in Figure 15.2, vertices a and b both have degree 1. An isolated point is a vertex of degree 0. Identcode Generation In Java Using Barcode creator for Java Control to generate, create Identcode image in Java applications. Barcode Reader In C# Using Barcode decoder for VS .NET Control to read, scan read, scan image in VS .NET applications. Figure 15.2 Subgraphs
UPC  13 Generation In .NET Framework Using Barcode creation for VS .NET Control to generate, create EAN13 image in .NET applications. Barcode Scanner In C#.NET Using Barcode Control SDK for .NET Control to generate, create, read, scan barcode image in .NET applications. Theorem 15.1 The sum of the degrees of the vertices of a graph with m edges is 2m. Each edge contributes 1 to the degree of each of the two vertices that determine it. So the total contribution if m edges is 2m. A complete graph is a simple graph in which every pair of vertices is connected by an edge. For a given number n of vertices, there is only one complete graph of that size, so we refer to the complete graph on a given vertex set. EXAMPLE 15.3 The Complete Graph on a Set of Four Vertices GS1  13 Generation In VS .NET Using Barcode encoder for Reporting Service Control to generate, create GS1  13 image in Reporting Service applications. Create EAN13 In ObjectiveC Using Barcode generator for iPad Control to generate, create EAN / UCC  13 image in iPad applications. Figure 15.3 shows the complete graph on the set V = {a, b, c, d}. Its edge set E is E = {ab, ac, ad, bc, bd, cd}. Note that the graphs in the previous examples are subgraphs of this one. Code 3/9 Drawer In VS .NET Using Barcode generator for Visual Studio .NET Control to generate, create Code 39 Extended image in .NET applications. Barcode Decoder In .NET Using Barcode reader for VS .NET Control to read, scan read, scan image in .NET applications. Figure 15.3 A complete
Theorem 15.2 The number of edges in the complete graph on n vertices is n(n 1)/2. There are n vertices, and each of them could be adjacent to n 1 other vertices. So there are n(n 1) ordered pairs of vertices. Therefore, there are n(n 1)/2 unordered pairs because each unordered pair could be ordered in two ways. For example, the unordered pair {a, b} can be ordered as either (a, b) or (b, a). For example, the number of edges in the complete graph on the fourvertex set in Example 15.3 is n(n 1)/2 = 4(4 1)/2 = 6. Corollary 15.1 The number of edges in any graph on n vertices is m PATHS AND CYCLES A walk from vertex a to vertex b in a graph is a sequence of edges (a0 a1, a1 a2, . . . , ak 1 ak) where a0 = a and ak = b, that is, a sequence of edges (e1, e2, . . . , ek) where if edge ei connects some vertex to vertex ai then the next edge ei+1 connects that vertex ai to some other vertex, thereby forming a chain of connected vertices from a to b. The length of a walk is the number k of edges that form the walk. n(n 1)/2. CHAP. 15]

