A Die of Another Color by Gardner M.

By Gardner M.

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There is an edge [p, 41 with vertex p in set R and vertex 4 in set C iff b,, = 1. Thus z is the total number of nonzero elements in B. The labeled row graph R, and the labeled column graph R, are, respectively, the labeled graphs of B * B’ and B‘ * B ; where * denotes that in computing the inner product of vectors in the above matrix multiplications only Boolean addition 0 has been used, namely, 1 0 1 = 1. Since B * B’ and B’ * Bare both symmetric, therefore zR (the number of edges in 0,)and T, (the number of edges in 0,)are equal to the total number of ones that lie above the diagonals of B * B’ and B’ * B, respectively.

4) ek’(Lkp) = i < k; pi? qik)Pk, ei‘(Lkp)= qlk)pk+ pi, i > k. Thus Lkp can be obtained from p by replacing its kth element by zero and then adding p,q(,) to it. 2) is used. The column vector formed at each step is used in the next step as follows : A - ln = U , ( . . (U,(L, * f . (L2(L1n)). ). 5). The row vector formed at each step is used in the next step as follows : nA- ’ = (. ((nU,)U,). . U,)L,). . L1). Therefore, only these elements (along with the relevant bookkeeping information) are stored.

When A is positive definite),then the symmetry is preserved during the elimination process and the lower triangular part is not stored. 1), Aij = 0, for all i # j, except i = p o r j = p. 1). In order to discuss these methods, we will need some simple ideas from graph theory which are given in the next section. For additional details the reader is referred to Busacker and Saaty (1965), Harary (1969), Bellman et al. (1970). 4. Matrices and Graphs Let bij denote the ( i , j ) element of a matrix B which is obtained by replacing each nonzero element of A by unity.

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