By Lowell W. Beineke, Robin J. Wilson

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**Extra resources for Applications of graph theory**

**Sample text**

X Gr, and the connectivities still add: k = /Cj + k 2 + . . + Kr, and X = X1 + X2 + . . + Xr. It has further been proved (see [7]) that a product of optimal graphs also has optimal connectivity, in the sense defined in Section 4. The significance of these results is that we can readily define a class of configurations having some structural regularity and a specified connectivity, using no more edges than are necessary to attain that connectivity between the given number of vertices. The properties of small graphs are easily established by inspection: this is not so for large graphs in general, but if the large graph is a product then some of its properties follow from those of the constituent subgraphs.

In design it is necessary to identify the set of possible paths, and in operation one must select from the set a path which is available for use at the time of the demand. Dimensioning. How much equipment of each type should be provided? The simplest exchange with the structure raises the problem: what is the appropriate number s of cord circuits for a given set of r user stations? The only obvious bounds fors are 1 With maximum provision there will always be a cord circuit available on demand, whereas with a very small number there is a high probability that all will be busy, and that fresh calls will be blocked.

Alternatively, we can construct a graph which comprises the com plete set of inlets V1 and outlets Vt , together with one path between each inlet—outlet pair: this will here be called the terminal graph. Figure 20 shows a terminal graph T derived by applying the last four operations of Fig. 19 (that is, all the open parallels) to a unit graph. Applying the closed-parallel operation (which when applied to the unit graph yields the channel graph Q yields the final structure. The latter can be considered as a product of the terminal graph T with the channel graph C.

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

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