What is a proper graph Laplacian? An operator-theoretic framework for graph diffusion
We introduce an operator-theoretic definition of a proper graph Laplacian as any matrix associated with a given graph that can be expressed as the composition of a divergence and a gradient operator, with the gradient acting between graph-related spaces and annihilating constant functions. This prov...
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| Tipo de recurso: | artículo |
| Estado: | Versión publicada |
| Fecha de publicación: | 2025 |
| País: | España |
| Institución: | Consejo Superior de Investigaciones Científicas (CSIC) |
| Repositorio: | DIGITAL.CSIC. Repositorio Institucional del CSIC |
| OAI Identifier: | oai:digital.csic.es:10261/422785 |
| Acceso en línea: | http://hdl.handle.net/10261/422785 https://api.elsevier.com/content/abstract/scopus_id/105027333928 |
| Access Level: | acceso abierto |
| Palabra clave: | Schur complement Diffusion operator Gradient-divergence formulation Graph Laplacian Magnetic Laplacian Operator theory Pseudoinverse |
| Sumario: | We introduce an operator-theoretic definition of a proper graph Laplacian as any matrix associated with a given graph that can be expressed as the composition of a divergence and a gradient operator, with the gradient acting between graph-related spaces and annihilating constant functions. This provides a unified framework for determining whether a matrix represents a genuine diffusive operator on a graph. Within this framework, we prove that the standard Laplacian, the fractional Laplacian, the d-path Laplacians, and the degree-attracting and degree-repelling Laplacians are all proper diffusive Laplacians. In contrast, the in-degree and out-degree Laplacians correspond to advection operators, while the signed, signless, magnetic, and deformed Laplacians are improper, as they cannot be written as the composition of a divergence and a true gradient. The magnetic Laplacian is shown to arise as the Schur complement of an extended proper Laplacian defined on a higher-dimensional space, a property also inherited by the signless Laplacian. The Lerman-Ghosh Laplacian is identified as a nonconservative diffusive operator coupled to an external reservoir. Finally, we prove that the Moore-Penrose pseudoinverse of the Laplacian is itself a proper Laplacian. This classification establishes a rigorous operator-theoretic foundation for distinguishing proper, nonconservative, and improper graph Laplacians. |
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