A^'-TERMINAL SWITCHING CIRCUITS 



681 



Number of Graphs 



Let G(iV, K) be the number of topologically distinct linear graphs which 

 can be drawn interconnecting the iV-terminals and using K branches. 

 G{N, K) counts all graphs including graphs with dangling branches and 

 disconnected pieces. It also counts graphs in which any or all of the N- 

 terminals are connected to no branches. Figures 7a, b, c, d, e show some 

 topologically distinct graphs which would be counted in finding G(3, 10). 



4 6 8 10 20 40 60 80100 200 



NUMBER OF TERMINALS, N 



Fig. 6 — Number of possible states of iV terminals. 



400 600 1000 



Graph 7f is topologically identical with graph 7b and so is not to be counted 

 again. The first step toward finding a lower bound on the number of con- 

 tacts which almost all switching functions require is to find an upper bound 

 onG(i\^, i^). 

 Theorem VI. 



G(N, K) < l^'-^^iN + 2K)^ . 



