## Download Applications of graph theory by Lowell W. Beineke, Robin J. Wilson PDF

By Lowell W. Beineke, Robin J. Wilson

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**Example text**

W. CATTERMOLE 3. Elementary Graphs An obvious method of interconnecting a few stations is to provide a link from each one to every other; this gives a mesh, or complete graph Kn. Clearly this is not practicable for (say) 104 stations, which would need about 108 links. At the other extreme, we can try to connect n vertices with a minimal graph —namely, a tree with n - 1 edges. There are many different trees —indeed a classical problem of graph theory was to enumerate them —but in our present context the star graph is the most important.

3 It should be said that there are many other graph and matrix repre sentations of connecting networks; however, we shall restrict our attention to those described here. Diagrams such as Fig. 1, which are commonly used to represent the network in the large, tend to conceal what every telephone engineer knows —that within each switching center represented by a dot on the grand plan, there is an internal connecting network of no less com plexity. A telephone exchange contains a large number of switches, at least some tens of thousands of contacts arranged in some hundreds 20 K.

Permutations can be classed according to the number A,- of cycles of length i. The cycle index of a permutation group T is a polynomial in variables x (- (as many as the degree of the group), P(G:xl, x 2, .. ) = |p | ^ 7 TT x i ‘> where the summation is taken over all permutations of the group T, 48 K. W. CATTERMOLE Identity Reflection Vertex permutation (1)(2)(3)(4) Rotation through 90° (12)(34) Edge permutation (a)(b)(c)(d)(e)(J) (a)(c)(bd)(ef) (abcd)(ef) Fig. 30 and Irl is the order of the group —that is, the number of distinct per mutations.