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

By Lowell W. Beineke, Robin J. Wilson

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L. Hakimi, Graphs with given connectivity and indepen­ dence number or networks with given measures of vulnerability and surviv­ ability, IEEE Trans. Circuit Theory CT-20 (1973), 2—10; MR 48 #10675. 3. D. Bear, “ Principles of Telecommunication Traffic Engineering” , Peter Peregrinus, London (1976). 4. V. E. Benes, “ Mathematical Theory of Connecting Networks and Telephone Traffic” , Academic Press, New York and London (1965); MR 35 #1408. 5. C. Berge, “ Principles of Combinatorics” , Academic Press, New York (1971); MR 38 #5635 and 42 #5805.

15(b)) clearly has connectivity c. When a connection is demanded, the probability that a path is avail­ able increases with c, other things being equal. Clos [13] showed that with a sufficiently large value of c this probability becomes 1 —that is, the network is non-blocking. The criterion for this is that - 1, where b is the number of inlets to one peripheral {A or C) switch. The reasoning is that at most b - 1 of the c pa other connections to the same A-switch, and a further - 1 by other connections to the same C-switch; with 2b —1 paths, at least one must be free.

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. In simple net­ works, many paths will be obvious on inspection; for example, the graph in Fig. 8 has four paths between vertices 1 and 4, as indicated. Generally, however, one would like to have a systematic method of enumerating these paths.

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