Discrete Mathematics with Graph Theory (3rd Edition) 291

Discrete Mathematics with Graph Theory (3rd Edition) 291 -...

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Chapter 10 289 (b) Dirac's Theorem says that a graph with n vertices is Hamiltonian provided the degree of every vertex is at least ~. This must be the case with the graph in question. To see this, assume deg v < 12° = 5 for some v E V. This implies that deg w :::; 8 for five other vertices w in V (those not adjacent to v). SO EVEV deg v < 5 + 5(8) + 4(9) = 81, again contradicting E VEV deg v = 21£1 = 82. 0 1 0 1 0 1 1 0 0 0 1 0 16. (a) A = 0 0 0 0 1 1 1 0 0 0 0 0 0 1 1 0 0 0 1 0 1 0 0 0 (b) The (1,5) entry of A2 is 1, because there is precisely one walk of length two from V1 to V5. Similarly, the (1,5) entry of A3 is one (V1 V6V3V5 is the unique walk of length three), the (1,5) en- try of A4 is 5 (V1V2V5V3V5, V1V2V5V2V5, V1V2V1V2V5, V1V4V1V2V5, V1V6V1V2V5), and the (1,5) entry of A5 is 6 (V1V2V1V6V3V5, V1V4V1V6V3V5, V1V6V1V6V3V5, V1V6V3V5V2V5, V1V6V3V5V3V5,
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Unformatted text preview: V1 V6V3V6V3V5). (c) Here is one possibility: cp(V1) = W5, cp(V2) = W3, cp(V3) = Wl. cp(V4) = W6, cp(V5) = W2, cp(V6) = W4 0 0 1 0 0 0 0 0 0 0 1 0 (d) P= 0 1 0 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 1 0 0 0 1 0 1 0 0 1 0 1 0 0 0 The adjacency matrix for '}{ is B = 0 1 0 0 1 0 1 0 0 0 1 0 0 0 1 1 0 1 0 0 0 0 1 0 We note that 0 0 1 0 0 0 0 1 0 1 0 1 0 0 0 0 1 0 0 0 0 0 1 0 1 0 0 0 1 0 0 0 1 0 0 0 PAp T = 0 1 0 0 0 0 0 0 0 0 1 1 1 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 1 1 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 1 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 1 0 0 0 0 1 0 0 1 1 0 0 0 0 0 1 0 0 0 1 0 0 0 1 0 1 0 0 0 0 0 =B. 1 0 1 0 0 0 0 0 0 0 0 1 0 1 0 1 0 1 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 17. (a) The answer is 0 since there is no edge from V1 to V5....
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