观点

Proposition 10. Let G be a finite positive-vertex-weighted graph satisfying the cross-mass condition with constant C≥1 , and suppose that _vw(N_G(v))≤e^x for some x>X(C) . There is a finite positive-vertex-weighted graph G' with a …

Proposition 10. Let G be a finite positive-vertex-weighted graph satisfying the cross-mass condition with constant C≥1 , and suppose thatvw(NG(v))≤ex for some x>X(C) . There is a finite positive-vertex-weighted graph G' with a weight-preserving map to G that is injective on directed edges, such thatvw(NG' (v)), ≤e√x ,; TG' (w), ≥ (1-aC x√x)TG(w) -MG(w)x. It also satisfies the same cross-mass condition, and MG' (w)≤MG(w) .

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发起推演1

证明
Lemma 5 ([Triangle normalization and peeling]). For a positive-weighted finite…Lemma 9. There is a choice of these labels for which the projection (u,y)↦u…Lemma 4 ([Preservation under splitting]). Let (G,w) satisfy with constant C .…观点
Peel edges whose current common-neighbor weight is less than 1/x . If H is the remaining graph, Lemma 5 ([Triangle normalization and peeling]) gives T_H(w)≥T_G(w)-M_G(w)/x. Edge deletion does not increase any weighted neighborhood, and it preserves the cross-mass condition. If H has no edges, take G' to be empty. Then T_G(w)≤M_G(w)/x , which implies . Otherwise, apply Lemma 9 to H . Since x↦ x/√x is decreasing for x≥e^2 , ensures 0≤1-a_C x/√x≤1 . Hence T_ G' (w) , ≥(1-a_C( x)/(√x)) (T_G(w)-M_G(w)x); , ≥(1-a_C( x)/(√x))T_G(w) -M_G(w)x. The projection through H is still injective on directed edges. The edge-mass bound and inherited cross-mass condition follow from Lemma 4 ([Preservation under splitting]).
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