DefinitionDefinition: Literature-sourcedProof: Not applicablePipeline-generatedjudge pass (gpt-6-luna)

How statement and proof provenance work

The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

These labels describe origin, not correctness: citations and verification chips remain separate evidence.

Finite weighted graph Laplacian

Definition

Let $n \geq 1$ , $V = {1, \dots, n}$ , and let $W = (w_{i j}) \in R^{n \times n}$ satisfy $w_{i j} = w_{j i} \geq 0$ and $w_{i i} = 0$ . Define $d_{i} = \sum_{j = 1}^{n} w_{i j}$ , $D = diag (d_{1}, \dots, d_{n})$ , and $L = D - W$ . The support graph has vertex set $V$ and edge ${i, j}$ exactly when $w_{i j} > 0$ ; isolated vertices are singleton connected components. Put $V_{+} = {i \in V : d_{i} > 0}$ , let $D_{+}$ and $L_{+}$ be the principal submatrices indexed by $V_{+}$ , and, only when $V_{+} \neq \emptyset$ , define $L_{s y m, +} = D_{+}^{- 1 / 2} L_{+} D_{+}^{- 1 / 2}$ on $R^{V_{+}}$ .

Definition references

Ulrike von Luxburg, A Tutorial on Spectral Clustering, Statistics and Computing 17(4), 2007

Jean Gallier, Notes on Elementary Spectral Graph Theory: Applications to Graph Clustering Using Normalized Cuts, 2013

Used by

Dependency tree · 0 levels

Nothing. This result depends on no other item in the library.

Sources

Ulrike von Luxburg, A Tutorial on Spectral Clustering, Statistics and Computing 17(4), 2007 (standard reference, not scraped)
Jean Gallier, Notes on Elementary Spectral Graph Theory: Applications to Graph Clustering Using Normalized Cuts, 2013 (standard reference, not scraped)