We propose a Laplacian based on general inner product spaces, which we call the inner product Laplacian. We show the combinatorial and normalized graph Laplacians, as well as other Laplacians for hypergraphs and directed graphs, are special cases of the inner product Laplacian. After developing the necessary basic theory for the inner product Laplacian, we establish generalized analogs of key isoperimetric inequalities, including the Cheeger inequality and expander mixing lemma. Dirichlet and Neumann subgraph eigenvalues may also be recovered as appropriate limit points of a sequence of inner product Laplacians. In addition to suggesting a new context through which to examine existing Laplacians, this generalized framework is also flexible in applications: through choice of an inner product on the vertices and edges of a graph, the inner product Laplacian naturally encodes both combinatorial structure and domain-knowledge.

翻译：我们提出了一种基于一般内积空间的拉普拉斯算子，称为内积拉普拉斯算子。我们证明组合图拉普拉斯算子与归一化图拉普拉斯算子，以及超图和有向图的其他拉普拉斯算子，都是内积拉普拉斯算子的特例。在建立了内积拉普拉斯算子的必要基础理论后，我们构建了关键等周不等式（包括Cheeger不等式和扩展图混合引理）的广义类比。狄利克雷和诺伊曼子图特征值也可通过内积拉普拉斯算子序列的适当极限点来恢复。除了为审视现有拉普拉斯算子提供新视角外，该广义框架在应用中也具有灵活性：通过对图的顶点和边选择内积，内积拉普拉斯算子能自然地同时编码组合结构与领域知识。