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

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.

An $(n, d, λ)$ -graph and an expander

Definition

Let $G$ be a finite simple graph on $n \geq 2$ vertices. We call $G$ an $(n, d, λ)$ -graph when:

$G$ is $d$ -regular;
the adjacency eigenvalues satisfy $d = λ_{1} (G) \geq λ_{2} (G) \geq \dots \geq λ_{n} (G);$
the second-largest eigenvalue obeys $λ_{2} (G) \leq λ$ .

Thus $d - λ$ is a lower bound for the actual top spectral gap $d - λ_{2} (G)$ .

In the present page's elementary sense, an expander is a graph that is an $(n, d, λ)$ -graph with $λ < d$ .

Depends on

Adjacency spectrum, spectral radius, and cospectral graphs

Used by

For an (n,d,λ)-graph with λ<d, every nontrivial cut has many crossing edges Theorem

Dependency tree · two levels

5 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

O. Pikhurko, Algebraic Methods in Combinatorics, Chapter 14 (standard reference, not scraped)

An (n,d,λ)(n,d,λ)-graph and an expander