In graph theory, an adjacency matrix , A, is a way of representing which nodes (or vertices) are connected. For a simple directed graph, each entry, , is either 1 (if a direct path exists from node i to node j) or 0 (if no direct path exists from node i to node j). For example, consider the following graph and corresponding adjacency matrix. The entry is 1 because a direct path exists from node 1 to node 4. However, the entry is 0 because no path exists from node 4 to node 1. The entry is 1 because a direct path exists from node 3 to itself. The matrix indicates the number of ways to get from node i to node j within k moves (steps). Website Map A content map can be used to show how different pages on a website are connected. For example, the following content map shows the relationship among the five pages of a certain website with links between pages represented by arrows. The content map can be represented by a 5 by 5 adjacency matrix where each entry, a i j , is either 1 (if a link exists from page i to page j ) or 0 (if no link exists from page i to page j ). (a) Write the 5 by 5 adjacency matrix that represents the given content map. (b) Explain the significance of the entries on the main diagonal in your result from part (a). (c) Find and interpret A 2 .
In graph theory, an adjacency matrix , A, is a way of representing which nodes (or vertices) are connected. For a simple directed graph, each entry, , is either 1 (if a direct path exists from node i to node j) or 0 (if no direct path exists from node i to node j). For example, consider the following graph and corresponding adjacency matrix. The entry is 1 because a direct path exists from node 1 to node 4. However, the entry is 0 because no path exists from node 4 to node 1. The entry is 1 because a direct path exists from node 3 to itself. The matrix indicates the number of ways to get from node i to node j within k moves (steps). Website Map A content map can be used to show how different pages on a website are connected. For example, the following content map shows the relationship among the five pages of a certain website with links between pages represented by arrows. The content map can be represented by a 5 by 5 adjacency matrix where each entry, a i j , is either 1 (if a link exists from page i to page j ) or 0 (if no link exists from page i to page j ). (a) Write the 5 by 5 adjacency matrix that represents the given content map. (b) Explain the significance of the entries on the main diagonal in your result from part (a). (c) Find and interpret A 2 .
Solution Summary: The author calculates the Adjacency Matrix (A) for a content map, to explain the significance of the main diagonal and to find & interpret A 2.
In graph theory, an adjacency matrix, A, is a way of representing which nodes (or vertices) are connected. For a simple directed graph, each entry, , is either 1 (if a direct path exists from node i to node j) or 0 (if no direct path exists from node i to node j). For example, consider the following graph and corresponding adjacency matrix. The entry is 1 because a direct path exists from node 1 to node 4. However, the entry is 0 because no path exists from node 4 to node 1. The entry is 1 because a direct path exists from node 3 to itself. The matrix indicates the number of ways to get from node i to node j within k moves (steps).
Website Map A content map can be used to show how different pages on a website are connected. For example, the following content map shows the relationship among the five pages of a certain website with links between pages represented by arrows. The content map can be represented by a 5 by 5 adjacency matrix where each entry,
, is either 1 (if a link exists from page i to page j) or 0 (if no link exists from page i to page j).
(a) Write the 5 by 5 adjacency matrix that represents the given content map.
(b) Explain the significance of the entries on the main diagonal in your result from part (a).
rmine the immediate settlement for points A and B shown in
figure below knowing that Aq,-200kN/m², E-20000kN/m², u=0.5, Depth
of foundation (DF-0), thickness of layer below footing (H)=20m.
4m
B
2m
2m
A
2m
+
2m
4m
sy = f(x)
+
+
+
+
+
+
+
+
+
X
3
4
5
7
8
9
The function of shown in the figure is continuous on the closed interval [0, 9] and differentiable on the open
interval (0, 9). Which of the following points satisfies conclusions of both the Intermediate Value Theorem
and the Mean Value Theorem for f on the closed interval [0, 9] ?
(A
A
B
B
C
D
=
Q6 What will be the allowable bearing capacity of sand having p = 37° and ydry
19 kN/m³ for (i) 1.5 m strip foundation (ii) 1.5 m x 1.5 m square footing and
(iii)1.5m x 2m rectangular footing. The footings are placed at a depth of 1.5 m
below ground level. Assume F, = 2.5. Use Terzaghi's equations.
0
Ne
Na
Ny
35 57.8 41.4 42.4
40 95.7 81.3 100.4
Chapter 11 Solutions
Precalculus Enhanced with Graphing Utilities (7th Edition)
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