LemmaStatement: AI-generatedProof: AI-generated

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.

Weighted spectral bounds control the degree without loops

Statement

Let $A$ be a network on a finite nonempty $V$ , $n = ∣ V ∣$ , $d_{0} > 0$ , and $L = D - A$ with $D_{x x} = d_{A} (x) = \sum_{y} A_{x y}$ . Put $J_{x y} = 1$ , let $I$ be the identity matrix, and set $Δ_{L} = L - d_{0} I + (d_{0} / n) J$ . Suppose $c^{-} d_{0} I ⪯ Δ_{L} ⪯ c^{+} d_{0} I$ , where matrix order is the quadratic-form order on the vertex space. Then the degree without the loop, $s (x) = \sum_{y \neq x} A_{x y} = d_{A} (x) - A_{x x}$ , satisfies $(1 + c^{-}) d_{0} - d_{0} / n \leq s (x) \leq (1 + c^{+}) d_{0} - d_{0} / n$ for every $x$ . The network, vertex-space and matrix-order definitions are Finite symmetric networks and loop removal, Finite vertex Hilbert space in the article convention and Self-adjoint order, norm bounds and diagonal tests.

Facts & Assumptions

[F1]

The network Laplacian has diagonal entry $d_{A} (x) - A_{x x} = s (x)$ (Finite symmetric networks and loop removal).

[F2]

The vertex-space matrix action and inner product use finite coordinate sums (Finite vertex Hilbert space in the article convention).

[F3]

If $a I ⪯ B ⪯ b I$ for a Hermitian matrix, then $a \leq B_{x x} \leq b$ (Self-adjoint order, norm bounds and diagonal tests).

Proof

Given: The data in the Statement.

Goal: Establish the stated conclusions.

1.1F1F2algebra

The matrix $Δ_{L}$ is real symmetric, hence Hermitian in the vertex-space convention. Its diagonal entry equals $s (x) - d_{0} + d_{0} / n$ , since $I_{x x} = J_{x x} = 1$ .

2.1F3step 1.1algebra∎

Apply the diagonal order test [F3] to obtain $c^{-} d_{0} \leq s (x) - d_{0} + d_{0} / n \leq c^{+} d_{0}$ . Adding $d_{0} - d_{0} / n$ proves the pointwise bounds for every vertex. No upper bound on the loop weights was used.

Depends on

Used by · 2

Dependency tree · 2 levels

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