barcode in ssrs 2008 Figure 15.1 A graph in Java

Encoder EAN / UCC - 13 in Java Figure 15.1 A graph

Figure 15.1 A graph
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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
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EXAMPLE 15.2 Subgraphs
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GRAPHS
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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.
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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.
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Figure 15.2 Subgraphs
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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
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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.
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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 four-vertex 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.
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