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Solution of large, sparse linear systems is an important problem in science and engineering. Such systemsarise in many applications, such as electrical networks, stress analysis, and more generally, in the numeri-cal solution of partial differential equations. When the coefficient matrices associated with these linearsystems are symmetric and positive definite, the systems are often solved iteratively using the precondi-tioned conjugate gradient method. We have developed a new class of preconditioners, which we call sup-port tree preconditioners, that are based on the combinatorial properties of the graphs corresponding to thecoefficient matrices of the linear systems. We call the resulting iterative method support tree conjugategradient, or STCG. These new preconditioners are applicable to the class of symmetric and diagonallydominant matrices, and have the advantage of being well-structured for parallel implementation, both inconstruction and in evaluation. In this thesis, we present the intuition, construction, implementation, andanalysis of STCG.STCG is based upon an interesting isomorphism between a certain class of matrices (which we call Lapla-cian matrices), edge-weighted undirected graphs, and resistive networks. Using this isomorphism, weshow that an iterative method can be interpreted in terms of these discrete structures. Based on this inter-pretation, the STCG method for accelerating convergence is developed, which involves constructing other,more efficient discrete structures called support trees, and using their interpretation as matrices to applythem as preconditioners. Interestingly, the matrix preconditioners used in STCG are larger, but sparserthan conventional preconditioners. Additionally, the construction of support trees is basically an applica-tion of recursive divide-and-conquer. Support trees have very regular structures and are very well-suitedfor parallel implementation.Through theoretical analysis and numerical experiments, we show that STCG is practical and efficient forthe parallel solution of large sparse linear systems with Laplacian coefficient matrices. STCG is an inter-esting example of combinatorial techniques being applied to solve an algebraic problem. These techniqueshave wider applicability than the acceleration of iterative techniques. We also demonstrate an applicationof these techniques to the more general problem of bounding eigenvalues.