False statementConstruction: AI-adaptedVerification: AI-generatedSession-authored (Fable 5 assisted)precheck passaudited 2026-08-28

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FALSE: positive second Laplacian eigenvalue characterises 2-connectivity

Statement

False claim. A finite simple graph has positive second Laplacian eigenvalue if and only if it is $2$ -connected.

Facts & Assumptions

Given: The path graph $P_{3}$ on vertices $1 - 2 - 3$ .

[L1]

A graph with positive algebraic connectivity is connected, and conversely (A finite simple graph is connected if and only if its algebraic connectivity is positive).

[F1]

The algebraic connectivity is the second-smallest Laplacian eigenvalue (The algebraic connectivity of a finite simple graph).

[F2]

A graph is $2$ -connected when deleting any one vertex leaves it connected (Vertex cuts, edge cuts, vertex connectivity $κ (G)$ and edge connectivity $λ (G)$ , with conventions for complete and one-vertex graphs).

Refutation

technique · direct

1.1

The path $P_{3}$ is connected, so [L1] and [F1] show that its second Laplacian eigenvalue is positive.

L1F1

1.2

Deleting the middle vertex of $P_{3}$ leaves two isolated vertices, which is disconnected. Therefore [F2] shows that $P_{3}$ is not $2$ -connected.

2.1

So $P_{3}$ has positive second Laplacian eigenvalue but is not $2$ -connected, refuting the claim.

step 1.1step 1.2∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

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Sources

O. Pikhurko, Algebraic Methods in Combinatorics, Section 14.2 (standard reference, not scraped)