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By Lowell W. Beineke, Robin J. Wilson

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Theory and Application of Graphs - download pdf or read online

Within the spectrum of arithmetic, graph concept which experiences a mathe­ matical constitution on a collection of components with a binary relation, as a famous self-discipline, is a relative newcomer. In contemporary 3 many years the fascinating and swiftly becoming zone of the topic abounds with new mathematical devel­ opments and demanding functions to real-world difficulties.

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There is an extensive literature, mostly derived from the pioneer work of Jacobaeus [32] which has been admirably summarized and extended by Elldin [16]; textbook treatments are also available (see, for example, [3] and [49]). The structural proper­ ties of switched networks were brought to the fore in the 1950s by the work of Clos [13] and others (see [4] and [36]). The most explicit development of /-stage graph theory for switched networks was initiated by Takagi [50] and [51 ], whose methods have been extended by other recent writers (see, for example, [12], [30], [52] [54] and [55]).

Of course we are still a long way from solving a blocking problem by such methods. The purpose of these remarks is to stimulate interest in the application to connection networks of combinatorial techniques which have been developed in other fields —developed a Logm (number of states) Fig. 32 2 GRAPH THEORY AND COMMUNICATIONS NETWORKS 51 good deal further, in fact, than this brief introduction shows. A further purpose is to suggest that, whereas the enumeration of states is obviously impracticable in a non-trivial network, the enumeration of equivalence classes may not be so.

For example, the graph shown as 3c in Table IV has six per­ mutations, and its four vertices have 24 permutations: so the number of members is 24/6 = 4, again in accordance with our enumeration. If the connecting network has the configuration of a complete graph Kn with one link per edge, then the problem is identical with that of counting unlabeled graphs (see [26] or [29] for a general solution, and [42]). In practice, connecting networks are usually in­ complete. An incomplete graph of sufficient regularity will however have a permutation group, and so is amenable to this approach.

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

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