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GRAPH THEORY
[CHAP. 8
Fig. 8-59
SPECIAL GRAPHS 8.48. Draw two 3-regular graphs with: (a) eight vertices; (b) nine vertices. 8.49. Consider the complete graph Kn . (a) Find the diameter of Kn . (b) Find the number m of edges in Kn . (c) Find the degree of each vertex in Kn . (d) Find those values of n for which Kn is: (i) traversable; (ii) regular. 8.50. Consider the complete graph Km,n . (a) Find the diameter of Km,n . (b) Find the number E of edges in Km,n . (c) Find those Km,n which are traversable. (d) Which of the graphs Km,n are isomorphic and which homeomorphic? 8.51. The n-cube, denoted by Qn , is the graph whose vertices are the 2n bit strings of length n, and where two vertices are adjacent if they differ in only one position. Figure 8-60(a) and (b) show the n-cubes Q2 and Q3 . (a) Find the diameter of Qn . (b) Find the number m of edges in Qn . (c) Find the degree of each vertex in Qn . (d) Find those values of n for which Qn is traversable. (e) Find a Hamiltonian circuit (called a Gray code) for (i) Q3 ; (ii) Q4 .
Fig. 8-60