nLab discrete Laplace operator

Idea

The discrete Laplace operator is a discrete analogue, defined on a graph, of the Laplace operator of a Riemannian manifold.

Intuitively it measures how much a function at a vertex differs from its average over the adjacent vertices.

Motivation

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Definitions

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For finite graphs

Normalized version

For weighted graphs

For directed graphs

Spectral properties

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See also

• graph
• Laplace operator
• harmonic function
• Markov chain