DefinitionDefinition: Literature-sourcedProof: Not applicableSession-authored (Fable 5 assisted)judge pass (gpt-5.6-terra)audited 2026-08-28

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The Laplacian matrix of a finite simple graph

Definition

Let $G$ be a finite simple graph with ordered vertex set $V (G) = {v_{i} : i < n} = {v_{0}, \dots, v_{n - 1}}$ , adjacency matrix $A (G)$ in that order, and degrees $d_{i} = \deg (v_{i})$ for $i < n$ . The diagonal matrix

$D (G) : = diag (d_{0}, \dots, d_{n - 1})$

is the degree matrix of $G$ , and

$L (G) : = D (G) - A (G)$

is the Laplacian matrix of $G$ .

Equivalently, the entries of $L (G)$ are

$L (G)_{i j} = {\begin{cases} \deg (v_{i}), & if i = j, \\ - 1, & if i \neq j and v_{i}, v_{j} are adjacent, \\ 0, & otherwise, \end{cases} (i, j < n) .$

Since $A (G)$ is symmetric and $D (G)$ is diagonal, $L (G)$ is symmetric.

Depends on

Used by

Dependency tree · two levels

6 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

Richard P. Stanley, MIT 18.314 handout, The Matrix-Tree Theorem (standard reference, not scraped)