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Figure 15.53 Incidence matrices
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Figure 15.54 Digraph
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Figure 15.55 Incidence matrix
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Figure 15.56 Digraph
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The adjacency matrix is shown in Figure 15.57. The adjacency list is shown in Figure 15.57. The graph is not connected because there is no path from B to A. The graph is not acyclic because it contains the cycle BECDB.
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Figure 15.57 Adjacency matrix and adjacency list
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a. The adjacency matrix for a wheel graph looks like matrix A shown in Figure 15.58 on page 316. b. The incidence matrix for a wheel graph looks like matrix B shown in Figure 15.58 (for the case n = 4). In general, it will have n 1s followed by n 0s on the first row. Below that will lie the identity matrix (all 1s on the diagonal and 0s elsewhere) followed by the square matrix with 1s on the diagonal and the subdiagonal. Compare this with the recursive solution to Problem 15.9 on page 308. c. The adjacency list for a wheel graph looks like the list shown in Figure 15.58 on page 316. The edge list for the first vertex (the central vertex) has n edge nodes, one for every other vertex. Every other edge list has three edge nodes: one pointing to the central vertex (labeled a in Figure 15.58) and one to each of its neighbors. a. The two graphs are isomorphic. The bijection is defined by the vertex labels shown in Figure 15.59 on page 316. b. An euler cycle for G2 is ABCDEBFCADFEA. c. A hamiltonian cycle for G2 is ABCDFEA. The trace of Dijkstra s algorithm is shown in Figure 15.60 on page 316. The trace of Dijkstra s algorithm is shown in Figure 15.61 on page 316. a. If the depth-first search is applied to a tree, it does a preorder traversal. b. If the breadth-first search is applied to a tree, it does a level-order traversal.
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15.15 15.16 15.17
GRAPHS
[CHAP. 15
Figure 15.58 Adjacency matrix and adjacency list
Figure 15.59 Isomorphic graphs
Figure 15.60 Dijkstra s algorithm
Figure 15.61 Dijkstra s algorithm
CHAP. 15]
GRAPHS
The seven graphs are labeled in Figure 15.62. Among them: G1 is isomorphic to G2 : The isomorphism is shown by the vertex labels a j. G3 is isomorphic to G4 : The isomorphism is shown by the vertex labels p y. G6 cannot be isomorphic to any of the other graphs because it has 25 edges and all the others have 20. G3 (and thus also G4) cannot be isomorphic to any of the other graphs because it has a pyramid of four adjacent 3-cycles (pqr, prs, pst, and ptq) and none of the other graphs (except G6) does. G6 cannot be isomorphic to any of the other graphs because it has a chain of three adjacent 4cycles (ABCD, CEFG, and FHIJ) and none of the other graphs (except G6) does. Similarly, G7 cannot be isomorphic to any of the other graphs because it has a chain of four adjacent 3-cycles (PQS, QSR, SRT, and RTU) and none of the other graphs (except G6) does.
Figure 15.62 Graph isomorphisms
15.19 15.20
The adjacency matrix and the adjacency list are shown in Figure 15.63 on page 318. a. The breadth-first search visits ABDECHFIGKLJMONPQ; its spanning tree is shown on the left in Figure 15.64 on page 318. b. The depth-first search visits ABCFEIHDKLMJGNPQO; its spanning tree is shown on the right in Figure 15.64 on page 318.
GRAPHS
[CHAP. 15
Figure 15.63 Adjacency matrix and adjacency list
Figure 15.64 Breadth-first search and depth-first search
APPENDIX
Essential Mathematics
This appendix summarizes mathematical topics used in the study of data structures. THE FLOOR AND CEILING FUNCTIONS The floor and ceiling functions return the two nearest integers of a given real number. The floor of x, denoted by x , is the greatest integer that is not greater than x. The ceiling of x, denoted by x , is the smallest integer that is not smaller than x. Here are the main properties of these two functions. (The symbol stands for the set of all integers.) Theorem A.1 Properties of the Floor and Ceiling Functions 1. x = max{m | m x}, and x = min{n | n x}. 2. x x < x + 1, and x 1 < x x . 3. x 1 < x x x < x + 1. 4. If n and n x < n + 1, then n = x . If n and n 1 < x < n, then n = x . 5. If x , then x = x = x . 6. If x , then x < x < x . 7. x x and x = x 8. x + 1 = x and x + 1 x LOGARITHMS The logarithm with base b of a positive number x is the exponent y on b for which b y = x. For example, the logarithm of 1000 base 10 is 3 because 103 = 1000. This is written log10 1000 = 3. The logarithm with base 2 is called the binary logarithm and is written lg x = log 2 x. For example, lg 8 = 3. As a mathematical function, the logarithm is the inverse of the exponential function with the same base: y = log b x by = x For example, 3 = lg 8 because 23 = 8.
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