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Figure 15.51 Hamiltonian cycle
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Figure 15.52 Nonisomorphic graphs
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The adjacency matrix for the complete graph on n vertices is an n-by-n boolean matrix with false value at each entry on the diagonal and true value at every other entry. The incidence matrices Mn for the complete graphs on n vertices are as follows: Its has n rows and n(n 1)/2 columns (see Theorem 15.2 on page 286). If n = 2, it is the 2-by-1 matrix containing true in both entries. If n > 2, it is the matrix A concatenated horizontally with the matrix obtained from Mn 1 by placing one row of all false values on top of it. The four matrices are shown in Figure 15.53. a. b. c. d. e. f. a. b. c. d. e. f. The digraph is shown in Figure 15.54. Yes, this is a digraph: It has at least one one-way edge. No, the digraph is not strongly connected: There is no path from C to D. Yes, the digraph is weakly connected: Its embedded (undirected) graph is connected. No, the digraph is not acyclic: It contains the cycle AFEDA. Its adjacency matrix is shown in Figure 15.55 on page 315. The digraph G1 is shown in Figure 15.56 on page 315. Yes it is a digraph: Its adjacency matrix is not symmetric. No, this is not a simple digraph because it has a loop. Yes, this digraph is strongly connected. Yes, this digraph is weakly connected. No, the digraph is not acyclic: It contains the cycle ADB.
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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
Figure 15.56 Digraph
a. b. c. d.
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.
Figure 15.57 Adjacency matrix and adjacency list
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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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
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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.
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