Computational problems in applied mathematics often have strong ties to graph/grid based structures. We treat graph-based problems in applied mathematics. We consider the spectral bisection problem for general graphs, and extend Miroslav Fiedler's results to more general graphs, using the theory of irreducible matrices. In addition, we apply the spectral bisection theory to graph partitioning, and create a cascadic Multigrid algorithm to solve for the Fiedler vector of a graph Laplacian, thus producing a bisection. We give convergence results for a sub class of graphs for our method. Finally, we consider the numerical computation of the fair price of barrier options. In practice, this becomes a problem of solving a partial differential equation numerically. We introduce a technique where the grid is treated in space and time simultaneously. We use a Multigrid V-cycle, in which the coarse grids are chosen adaptively, based on the properties of the finer structured graph.
