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Summary

The Laplacian matrix is a matrix representation of a graph used in the analysis of random walks and electrical networks on graphs. It is a discrete analog of the Laplacian operator in multivariable calculus, which measures to what extent a graph differs at one vertex from its values at nearby vertices.
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It is implemented in the Wolfram Language as KirchhoffMatrix.
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The Laplacian also appears in the matrix tree theorem.
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Summaries from the best pages on the web

Summary
In the mathematical field of graph theory , the Laplacian matrix , also called the graph Laplacian , admittance matrix , Kirchhoff matrix or discrete Laplacian , is a matrix representation of a graph

Laplacian matrix - Wikipedia
wikipedia.org

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csustan.edu

Constructing a Laplacian for graphs The Laplacian matrix \(L\) for a graph \(G := (V, E)\) captures the same idea as the Laplacian for continuous, multivariate ...

The graph Laplacian - Matthew N. Bernstein
mbernste.github.io

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upenn.edu

Summary
The Laplacian matrix is a discrete analog of the Laplacian operator in multivariable calculus, which serves a similar purpose by measuring to what extent a graph differs at one vertex from its values at nearby vertices. It is used in the analysis of random walks and electrical networks on graphs, and is implemented in the Wolfram Language as KirchhoffMatrix. The Laplacian also appears in the matrix tree theorem.

Laplacian Matrix -- from Wolfram MathWorld
wolfram.com

Unable to generate a short snippet for this page, sorry about that.

ucsd.edu

This MATLAB function returns the graph Laplacian matrix, L. L = laplacian( G ) returns the graph Laplacian matrix, L . Each diagonal entry, L(j,j) , is given ...

Graph Laplacian matrix - MATLAB laplacian
mathworks.com