. The Bell System technical journal . Fig. 7 — Example of a shortest spanning subtree of a labelled graph withsome lengths negative. ®o -^ 0®. Fig. 85(a). The longest spanning subtree of the labeled graph of Figs. 4(a) and 1398 THE BELL SYSTEM TECHNICAL JOURNAL, NOVEMBER 1957 The longest spanning subtree of a connected labelled graphalso maximizes all increasing symmetric functions, and mini-mizes all decreasing symmetric functions, of the edge lengths. For example, with positive lengths the same spanning subtree thatminimizes the sum of the edge lengths also minimizes the product andthe squar
. The Bell System technical journal . Fig. 7 — Example of a shortest spanning subtree of a labelled graph withsome lengths negative. ®o -^ 0®. Fig. 85(a). The longest spanning subtree of the labeled graph of Figs. 4(a) and 1398 THE BELL SYSTEM TECHNICAL JOURNAL, NOVEMBER 1957 The longest spanning subtree of a connected labelled graphalso maximizes all increasing symmetric functions, and mini-mizes all decreasing symmetric functions, of the edge lengths. For example, with positive lengths the same spanning subtree thatminimizes the sum of the edge lengths also minimizes the product andthe square root of the sum of the squares. At the same time, it maximizesthe sum of the reciprocals and the product of the arc cotangents. It seems likely that these extensions of the original class of problemssoluble by PI and P2 contain many examples of practical interest inquite different contexts from the original connection networks. A notentirely facetious example is the following: A message is to be passedto all members of a certain underground organization. Each memberknows some of the other members and has procedures for arranging arend
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