Which (finite, undirected) graphs have this property?
Every vertex $v$ can be labeled with a positive integer $l(v)$.
Variant 1: For each vertex $v$, $l(v) \geq \Sigma_{[v,w] \in E, w \neq v} l(w)/2$.
Variant 2: For each vertex $v$, $l(v) > \Sigma_{[v,w] \in E, w \neq v} l(w)/2$.