Before revision: The combinatorial graph Laplacian has been a fundamental object in the analysis of and optimization on graphs. Its spectral structure has been widely used in graph optimization problems (e.g, clustering), and it is also connected to network circuit theory. There is also a topological view of the graph Laplacian, which permits the extension of the graph Laplacian to the more general q-th combinatorial Laplacian \Delta_q^K for a simplicial complex K.
After revision: We present a thorough study of the theoretical properties and devise efficient algorithms for the extension of the graph Laplacian to the more general q-th combinatorial Laplacian \Delta_q^K for a simplicial complex K.