By Lowell W. Beineke, Robin J. Wilson

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**Extra info for Applications of graph theory**

**Sample text**

Connectivity Telecommunication networks commonly offer alternative paths between any nominated pair of terminals. This is necessary because any link is available to many sources of traffic, and so may constitute part of one connection at a time when another connection is demanded. Other things being equal, the more alternative paths are available, the lower is the probability that all are busy and the new call blocked. Let us consider a graph with alternative paths, and try to determine the number of paths between two nominated vertices.

21(a) Fig. 8 + 0(ps), for Fig. 21( and p2 1q0 + 52 p 3q9 + 305p4qs + 0(p5), for Fig. ) As a numerical example, if p - 0-1, then Pa = 0-0075 and Pb= 0-0113. On comparing the coefficients term by term it is seen that those of Pb are never less than those of Pa■some ' are s (not shown) are the same. Consequently, > P a for every p. On closer examination, a much stronger result emerges. We can partition the r busy links into r, from the first stage, from the second, and so on, with rx+ r2 + r3 +r4 = r.

T. Boesch and R. E. Thomas, On graphs of invulnerable communication nets, IEEE Trans. Circuit Theory CT-17(1970), 183-192; MR 42 #2865. 7. K. W.

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

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