The Obstacle Problem on Graphs and Other Results

Our primary motivation is existence and uniqueness for the obstacle problem on graphs. That is, we look for unique solutions to the problem $Lu = \chi_{\{u>0\}}$, where $L$ is the Laplacian matrix associated to a graph, and $u$ is a nonnegative real-valued vector with preassigned zero coordinates and positive coordinates to be determined. In the course of solving this problem, we make a detour into the study of Laplacian matrices themselves. First, we present the row reduced echelon form of such matrices and determine the invertibility of proper square submatrices. Next, we determine eigenvalues of several simple Laplacians. In this context, we introduce a new polynomial called the generalized characteristic polynomial that allows us to compute (theoretically, if inefficiently)the usual characteristic polynomial for trees by inspection of the graph. Finally, we give our solution to the obstacle problem on graphs and discuss other components of the obstacle problem, which we investigate in future research.