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A graph-theoretic study of the computational efficiency of the generalized loop analysis and the generalized cutset analysis is presented. It is shown that the choice of an optimum mode of analysis will give rise to the sparsest loop impedance matrix and the sparsest cutset admittance matrix, respectively. The problem of formulating efficient algorithms for determining the optimum choice is shown to be strictly a problem in nonoriented linear graph. Two algorithms based on the concept of basis graph are presented and illustrated in detail with examples. A nonplanar version of the mesh analysis which generally yields a rather sparse loop impedance matrix is also included.