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21) 1 where Npa, D," = 0, 1, or - 1. D," is called the primitive cutset matrix associated with T. An arbitrary cutset q is expressed, uniquely, as a linear combination of da's and iii is equal to the rank m of the graph and n - iii to the nullity k . The following theorem is obvious from the preceding considerations. 6. The number of branches of a tree on a graph is equal to the rank m of the graph, and that of branches of a cotree to the nullity k. 1. A loop-free subset of m branches is a tree, and a cutsetfree subset of k branches is a cotree.

23M-msuch that u u di M -m 8,= 8= (ij+nuu6-n,), (Binsj=@,i # j ) . 4) For a given graph 8 = {%, 8, (3+, d - } , we can construct M - m graphs 6 ,= {92,, 23,,d i + , (3,-} (i = I , . . , M - m), where 111, and Bi are as defined previously, and di+ and di- are the restrictions of 8' and 8- to %,, respectively. 8 , ' s are called the connected components of 6. 6 is said to be connected when M - m = 1 and unconnected when M - m 2 2. 1. , M - m ) . 5) 6 E V(G, 8) such that nu - np = 86, and vice versa.

A branch and its endnodes. An incidence relation can be represented by other functions. , the set of all subsets of 23): a+: 111+29, 6-: 111-+29, and which satisfy the conditions 6 + n a n 6+nB = 6-n, n 6 - n B = @ (empty set) for every pair na , nB and M U 6+na= a= 1 u M a= 1 6-na=B. (E 111, cx # p), (7. I) 30 111. Topological Properties of Networks Then we can define d + and S'n, 3 b, 6-n, 3 a- uniquely by the relations if and only if d+b, = n a , if and only if a-b, b, = n,. 2). S'n, and 6-n, are intuitively interpreted as the set of branches positively incident to norand the set of branches negatively incident to nor, respectively (see Fig.