Discrete Mathematics with Graph Theory (3rd Edition) 252

Discrete Mathematics with Graph Theory (3rd Edition) 252 -...

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250 Solutions to Exercises 6. (a) n-1 7. (b) For n = 1, the answer is O. For n ~ 2, the largest number of edges will occur when every pair of vertices is joined. The number of pairs of n vertices is (;). (c) For n = 1,0 is a graph with the maximum number of edges. For n ~ 2, IC n has the maximum number of edges. 8. (a) The graph is left to the student. To see that it is bipartite, associate with each vertex the color-red or black-of the square it represents and note that a knight always moves to a square of a different color. Thus, in the graph, each edge joins a red vertex and a black vertex. (The red vertices and the black vertices are the bipartition sets.) (b) The degrees of the vertices are shown to right inside the squares they represent. There are four ver- tices of degree 2, eight vertices of degree 3, twenty vertices of degree 4, sixteen vertices of degree 6 and sixteen vertices of degree 8. 9. (a) i. [BB] VI V2 V3 V4 V5 VI
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Unformatted text preview: 0 1 2 2 2 V2 1 0 1 2 3 V3 2 1 0 1 2 V4 2 2 1 0 2 V5 2 3 2 2 0 V6 1 2 1 1 1 ii. [BB] Vertex VI V2 column total 8 9 vertex degree 2 2 accessibility index 4 4.5 V6 1 2 1 1 1 0 2 3 4 4 4 4 3 2 3 4 6 6 6 6 4 3 4 6 8 8 8 8 6 4 4 6 8 8 8 8 6 4 4 6 8 8 8 8 6 4 4 6 8 8 8 8 6 4 3 4 6 6 6 6 4 3 2 3 4 4 4 4 3 2 V3 V4 V5 V6 7 8 10 6 3 2 1 4 2.3 4 10 1.5 The accessibility index is a measure of average path length from a given vertex to the oth-ers. If a vertex were connected to every other vertex, its accessibility index would be 1; the corresponding city would be as accessible as possible. In the present case, V6 has the lowest accessibility index; it is the most accessible. City V5 is least accessible. iii. [BB] Joining VI and V3 gives now eight edges, with still six vertices. . 8 . New beta mdex = '6 = 1.3. New accessibility indices are 2.3, 4.5, 1.5, 4, 10, 1.5. Cities V6 and V3 are now tied for most accessible; V5 is least accessible....
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