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.

Loop-dependent weighted degree envelope

Statement

Under the hypotheses of Weighted spectral bounds control the degree without loops, set $ℓ_{\min} = \min_{x} A_{x x}$ and $ℓ_{\max} = \max_{x} A_{x x}$ , with row degrees and loops as in Finite symmetric networks and loop removal. Define $d_{\min} = \min_{x \in V} d_{A} (x)$ and $d_{\max} = \max_{x \in V} d_{A} (x)$ . Then $(1 + c^{-}) d_{0} - d_{0} / n + ℓ_{\min} \leq d_{\min} \leq d_{\max} \leq (1 + c^{+}) d_{0} - d_{0} / n + ℓ_{\max}$ . More precisely, $s (x)$ may be replaced by $d_{A} (x) - A_{x x}$ in the pointwise bounds of that supplier. In particular $0 \leq A_{x x} \leq M$ implies $d_{\max} \leq (1 + c^{+}) d_{0} + M - d_{0} / n$ . When every loop vanishes the sharper upper bound is $(1 + c^{+}) d_{0} - d_{0} / n$ .

Facts & Assumptions

[F1]

The finite network has row degree $d_{A} (x) = s (x) + A_{x x}$ , with nonnegative loop weights (Finite symmetric networks and loop removal).

[F2]

The degree without loops satisfies $(1 + c^{-}) d_{0} - d_{0} / n \leq s (x) \leq (1 + c^{+}) d_{0} - d_{0} / n$ (Weighted spectral bounds control the degree without loops).

Proof

Given: The data in the Statement.

Goal: Establish the stated conclusions.

1.1F1F2algebra

Add $A_{x x}$ to both bounds in [F2] and use [F1]. Replacing $A_{x x}$ by $ℓ_{\min}$ in the lower bound and by $ℓ_{\max}$ in the upper bound gives a common interval containing every row degree. The finite nonempty vertex set ensures these loop extrema and the degree extrema exist.

2.1F1step 1.1algebra∎

Taking the minimum and maximum of the degrees in that interval gives the displayed chain. If $A_{x x} \leq M$ then $ℓ_{\max} \leq M$ , yielding the claimed uniform upper bound. If all loops vanish, $M = 0$ is allowed. This proves all assertions.

Depends on

Used by · 2

Dependency tree · 2 levels

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