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vb.net barcode printing Figure 15.9 shows the adjacency matrix for the graph in Figure 15.1 on page 285. in Java
Figure 15.9 shows the adjacency matrix for the graph in Figure 15.1 on page 285. Decode Data Matrix In Java Using Barcode Control SDK for Java Control to generate, create, read, scan barcode image in Java applications. Painting Data Matrix 2d Barcode In Java Using Barcode generator for Java Control to generate, create DataMatrix image in Java applications. Figure 15.9 An adjacency matrix
ECC200 Scanner In Java Using Barcode decoder for Java Control to read, scan read, scan image in Java applications. Bar Code Generator In Java Using Barcode encoder for Java Control to generate, create barcode image in Java applications. Note the following facts about adjacency matrices: 1. The matrix is symmetric, that is, a[i][j] == a[j][i] will be true for all i and j 2. The number of true entries is twice the number of edges. 3. Different orderings of the vertex set V will result in different adjacency matrices for the same graph. Adjacency matrices are often expressed with 0s and 1s instead of trues and falses. In that form, the adjacency matrix for Figure 15.1 would be the one shown in Figure 15.10 on page 291. Scanning Bar Code In Java Using Barcode decoder for Java Control to read, scan read, scan image in Java applications. ECC200 Generation In C#.NET Using Barcode encoder for .NET Control to generate, create Data Matrix 2d barcode image in .NET applications. CHAP. 15] Drawing Data Matrix 2d Barcode In .NET Using Barcode creator for ASP.NET Control to generate, create ECC200 image in ASP.NET applications. Drawing ECC200 In Visual Studio .NET Using Barcode encoder for VS .NET Control to generate, create ECC200 image in Visual Studio .NET applications. GRAPHS
Data Matrix 2d Barcode Maker In VB.NET Using Barcode printer for .NET framework Control to generate, create DataMatrix image in .NET applications. Code 128A Generator In Java Using Barcode printer for Java Control to generate, create Code 128 image in Java applications. THE INCIDENCE MATRIX FOR A GRAPH An incidence matrix for a graph (V, E) is a twodimensional array
Creating GS1128 In Java Using Barcode creator for Java Control to generate, create GS1128 image in Java applications. Encoding DataMatrix In Java Using Barcode drawer for Java Control to generate, create DataMatrix image in Java applications. int[][] a; Generating RoyalMail4SCC In Java Using Barcode creator for Java Control to generate, create British Royal Mail 4State Customer Barcode image in Java applications. Generate Code 128B In None Using Barcode printer for Microsoft Excel Control to generate, create Code 128 Code Set B image in Office Excel applications. obtained by ordering the vertices V = {v0 , v1, . . . , vn 1} and the edges E = {e0 , e1, . . . , em 1} and then assigning 1 to a[i][j] if vertex vi is incident upon edge ej and 0 otherwise. EXAMPLE 15.11 An Incidence Matrix ECC200 Drawer In VB.NET Using Barcode printer for VS .NET Control to generate, create Data Matrix image in .NET framework applications. Encode Code 128B In None Using Barcode printer for Office Word Control to generate, create Code128 image in Word applications. Figure 15.11 shows the incidence matrix for the graph in Figure 15.1 on page 285. The first row indicates that vertex a is incident upon edges 1, 2, and 3; the second row indicates that vertex b is incident upon edges 1 and 4, and so forth. Encode UPC Symbol In None Using Barcode printer for Online Control to generate, create Universal Product Code version A image in Online applications. Scan European Article Number 13 In Java Using Barcode reader for Java Control to read, scan read, scan image in Java applications. Figure 15.10 An adjacency matrix
Creating European Article Number 13 In None Using Barcode printer for Software Control to generate, create EAN13 Supplement 5 image in Software applications. Draw USS Code 39 In Visual Studio .NET Using Barcode maker for ASP.NET Control to generate, create Code39 image in ASP.NET applications. Note that for simple graphs, no Figure 15.11 A graph and its incidence matrix matter how many vertices and edges they have, there will always be exactly two 1s in each column of any incidence matrix. Why (See Review Question 15.9 on page 305.) THE ADJACENCY LIST FOR A GRAPH An adjacency list (or adjacency structure) for a graph (V, E) is a list that contains one element for each vertex in the graph and in which each vertex list element contains a list of the vertices that are adjacent to its vertex. The secondary list for each vertex is called its edge list. EXAMPLE 15.12 An Adjacency List Figure 15.12 shows the adjacency list for the graph in Figure 15.1 on page 285. The edge list for vertex a has three elements, one for each of the three edges that are incident with a; the edge list for vertex b has two elements, one for each of the two edges that are incident with b; and so on. Note that each edge list element corresponds to a unique 1 entry in the graph s corresponding incidence matrix. For example, the three elements in the edge list for vertex a correspond to the three 1s in the first row (the row for vertex a) in the incidence matrix in Figure 15.11. Figure 15.12 An adjacency list
GRAPHS
[CHAP. 15
Also note that the edge lists are not ordered, that is, their order is irrelevant. DIGRAPHS A digraph (or directed graph) is a pair G = (V, E) where V is a finite set and E is a set of ordered pairs of elements of V. As with (undirected) graphs, the elements of V are called vertices (or nodes) and the elements of E are called edges (or arcs). If e E, then e = (a, b) for some a, b V. In this case, we can denote e more simply as e = ab. We say that the edge e emanates from (or is incident from) vertex a and terminates at (or is incident to) vertex b. The outdegree of a vertex is the number of edges that emanate from it. The indegree of a vertex is the number of edges that terminate at it. Note that, unlike the graph definition, the digraph definition naturally allows an edge to terminate at the same vertex from which it emanates. Such an edge is called a loop. A simple digraph is a digraph that has no loops. EXAMPLE 15.13 A Digraph Figure 15.13 shows a digraph with vertex set V = {a, b, c, d} and edge set E = {ab, ad, bd, ca, dc}. Vertex a has outdegree 2 and indegree 1. Vertices b and c each have outdegree 1 and indegree 1. Vertex d has outdegree 1 and indegree 2.

