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 that _vw(N_G(v))≤e^x 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 that _vw(N_ G' (v)), ≤e^ √x ,; T_ G' (w), ≥ (1-a_C( x)/(√x))T_G(w) -M_G(w)x. It also satisfies the same cross-mass condition, and M_ G' (w)≤M_G(w) .