iV-TERMlNAL SWITCHING CIRCUITS 683 



Connect Bi to one of the A^-terminals or to one of the nodes just made or 

 else use Bi to make a new node. Connect B2 to one of the terminals or 

 nodes or else use B2 to make a new node, etc. The number of ways of con- 

 necting Biy • • • y Bk is less than 



{N-^K+ 1){N -i-K+l) '•' {N+2K)<{N+ 2Kf , 



which proves the theorem. 



Since most graphs can be constructed in many different ways by this 

 process, theorem VI gives a very poor estimate of G(iV, K). In the applica- 

 tion which we will make of G{Nj K) it is enough to know that log G{N, K) 

 behaves something like K log K. To prove that K log K cannot be replaced 

 by anything much smaller we now give a lower bound for G{Nj K). 



Theorem VII. 



Proof. G{Nf K) is larger than the number of graphs which can be drawn 

 without specifying certain nodes as terminals 1,2, • • • , iV. Of these graphs 

 let us count only those which have the property that no cycle in the graph 

 has an odd number of branches. Another characterization of these graphs 

 is that their nodes can be divided into two classes A and B such that no 

 branch joins two nodes of the same class. 



To construct such graphs we first number the branches 1,2, • • • ^ K and 

 give them an orientation (say by putting an arrow head at one end of each 

 branch). The front ends of the branches can be grouped together into nodes 

 in 4>{K) ways. Then the tail ends of the branches are grouped together in 

 one of <1>{K) ways. In this way a total of {<t>{K)y different graphs can be 

 drawn, in which the branches are numbered and oriented. If we now ignore 



the numbers on the branches we still have at least — /^ distinct graphs 



with oriented branches. If the orientation is ignored, the number of topo- 

 logically different graphs which remain is greater than 



Lower Bound 



We have seen that any switching function can be realized with no more 

 than about 



N^P2^ 



H - 2 log H 



