SHORTEST CONNECTION NETWORKS 



1397 



real numbers is the same as the negative of the minimum of the negatives 

 of the set. Therefore, PI and P2 can be used to construct a longest 

 spanning subtree by changing the signs of the "lengths" on the given 

 labelled graph. Fig. 8 gives, as an example, the longest spanning subtree 

 for the labelled graph of Figs. 4(a) and 5(a). 



It is easy to extend the arguments in support of NCI and NC2 from 

 the simple case of minimizing the sum to the more general problems of 

 minimizing an arbitrary increasing symmetric function, or of maximizing 

 an arbitrary decreasing symmetric function, of the edge "lengths" of a 

 spanning subtree. (A sjanmetric function of n variables is one whose 

 value is unchanged by any interchanges of the variable values; e.g., 

 .ri + X2 + • • • + .T„ , Xi Xo • • • Xn , sin 2.ri + sin 2.r2 + • • • + sin 2.r„ , 

 (.ri^ + x-i^ + • • • + Xn^y'~, etc.) From this follow the non-trivial generali- 

 zations. 



The shortest spanning siibtree of a connected labelled graph 

 also minimizes all increasing sjanmetric functions, and maxi- 

 mizes all decreasing symmetric functions, of the edge "lengths." 



(a) 



Fig. 7 — Example of a shortest spanning subtree of a labelled graph with 

 some "lengths" negative. 



®o -^ 0® 



Fig. 8 

 5(a). 



The longest spanning subtree of the labeled graph of Figs. 4(a) and 



