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There must be exactly two 1s in each column of an incidence matrix of a simple graph because each column represents a unique edge of the graph, and each edge is incident upon exactly two distinct vertices.
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15.1 a. b. c. d. e. f. g. h. i. j. k. n = 6. V = {a, b, c, d, e, f}. E = {ab, bc, bd, cd, ce, de, cf, df}. d(a) = 1, d(b) = 3, d(e) = d(f) = 2, d(c) = d(d) = 4. The path abcd has length 3. The path abcfde has length 5. The cycle bcedb has length 4. A spanning tree is shown in Figure 15.46. Its adjacency matrix is shown in Figure 15.47. Its incidence matrix is shown in Figure 15.47. Its adjacency list is shown in Figure 15.47.
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Figure 15.46 Spanning tree
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Figure 15.47 Adjacency matrix, incidence matrix, and adjacency list
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a. b. c. d. e. f. g. h. i. j. k.
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n = 6. V = {a, b, c, d, e, f}. E = {ad, ba, bd, cb, cd, ce, cf, de, ec, fe}. id(a) = id(b) = id(c) = id(f) = 1, id(d) = id(e) = 3. od(a) = od(d) = od(e) = od(f) = 1, od(b) = 2, od(c) = 4. The path adec has length 3. The path fecbad has length 5. The cycle adcba has length 4. A spanning tree is shown in Figure 15.48. Its adjacency matrix is shown in Figure 15.49. Its incidence matrix is shown in Figure 15.49.
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Figure 15.48 Spanning tree
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l. its adjacency list is shown in Figure 15.49.
Figure 15.49 Adjacency matrix, incidence matrix, and adjacency list
The complete graphs are shown in Figure 15.50:
Figure 15.50 Complete graphs
15.4 15.5 15.6
The graph G1 cannot be eulerian because it has odd degree vertices. But the hamiltonian cycle shown in Figure 15.51 on page 314 verifies that it is hamiltonian. The graph G2 is neither eulerian nor hamiltonian. a. b. c. d. e. f. g. h. i. j. k. l. Disconnected, cyclic, and spanning. Disconnected, acyclic, and spanning. Disconnected, cyclic, and spanning. Disconnected, cyclic, and not spanning. Connected, acyclic, and spanning. Connected, acyclic, and spanning. Connected, cyclic, and spanning. Connected, acyclic, and not spanning. Disconnected, cyclic, and spanning. Connected, cyclic, and not spanning. Disconnected, acyclic, and not spanning. Connected, acyclic, and not spanning.
The two graphs shown in Figure 15.52 on page 314 are not isomorphic because the one on the left has a 4-cycle containing two vertices of degree 2 and the one on the right does not. Yet, all five conditions of Theorem 15.4 on page 289 are satisfied.
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Figure 15.51 Hamiltonian cycle
Figure 15.52 Nonisomorphic graphs
15.8 15.9
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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