vb.net barcode printing Figure 15.9 shows the adjacency matrix for the graph in Figure 15.1 on page 285. in Java

Encode Data Matrix 2d barcode in Java Figure 15.9 shows the adjacency matrix for the graph in Figure 15.1 on page 285.

Figure 15.9 shows the adjacency matrix for the graph in Figure 15.1 on page 285.
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Figure 15.9 An adjacency matrix
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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.
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GRAPHS
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THE INCIDENCE MATRIX FOR A GRAPH An incidence matrix for a graph (V, E) is a twodimensional array
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int[][] a;
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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
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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.
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Figure 15.10 An adjacency matrix
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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
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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.
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