By Lowell W. Beineke, Robin J. Wilson

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An incomplete graph of sufficient regularity will however have a permutation group, and so is amenable to this approach. In some cases at least, a construction for the permutation group of a large graph may be known. Consider, for example, the cycle index of the Cartesian product of two permutation groups (see [5, p. 172]). Let T be a group acting on the set A, and let A be a group acting on the set B. The product group T x A acts on the set A x B\ its elements (7, 5) are defined by (7, 5)(a, b) = (7a, 8b).

The property of interest now is the connectivity between vertices in different sets: this is not necessarily the same as that within a set, but each can be found from the appropriate properties of the constituent subgraphs. For example, the connectivity of Kr r x Kr r#is 2/*, between two sets each of 2r2 vertices. 42 K. W. CATTERMOLE There are several product operations other than the Cartesian pro­ duct —for example, (a) the “lexicographic product” , or “ composition” [28, p. 21 ], whose con­ nectivity and related properties have been investigated in [201; (b) the “ Kronecker product” , constructed as follows: given a graph Gx with vertex-set {uu ..

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. 6. F. 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.

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

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