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By E.C. Titchmarsh

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Extra info for Eigenfunction expansions associated with second-order differential equations. Part 1

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21. The reason is that it would have to enter each vertex as many times at it leaves i t ; this is true even for the starting vertex. Hence there must be an even number of edges meeting at each vertex. Since this is not true for the graph in Fig. 21, no such path is possible. Evidently it is possible to pose this same problem for any graph, as did Euler himself. A path in a graph which traverses each edge exactly once is now called an Euler line or Euler circuit, and a graph in which an The Problem of the Konigsberg Bridges 37 Euler line exists is called an Euler graph.

Referring to Fig. 3, we see that starting from 0, the quickest route -goes either first to 1 or to 2, and thereafter follows the shortest route to 8. The justification for this statement is as before. 2 (6) where this notation, as before, means that we are to set j = 1 and j = 2 in succession in the expression in the parentheses, and to choose the smaller quantity. In the same way, the optimal, or best, route to 8 starting from 2 must go first to 0, 3 , or 4 (these being the only points directly accessible from 2), and must then follow the optimal route to 8.

Venttsel, “Navka,” Moscow, 1964. Assuming that equal distances require equal times to traverse, construct an array of travel times. in Moscow Map 8. Arbitrary Starting Point Having carried through the direct enumeration of possible routes for Fig. 3, and again for Fig. 11, we realize that although our method is simple in concept, it can be most tedious in application. Furthermore, we might want more information than we have heretofore asked. For example, we might wish to find the quickest path in Fig.

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