Before revision:  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. In this way, the standard graph Laplacian is simply the 0-th case (when q=0) of this family of operators. Taking this topological view, Wang et al. introduced the so-called persistent Laplacian \Delta_q^{K,L simplicial complexes K \hookrightarrow L . In this paper, we present a thorough study of properties and algorithms for persistent Laplacians.
After revision:  There is also a topological view of the graph Laplacian, which permits the extension of the graph Laplacian to the setting of pairs (or, in more generality, sequences) of simplicial complexes K \hookrightarrow L . In this paper, we present a thorough study of properties and algorithms for persistent Laplacians.