Given two graphs $G_{1}(E_{G1},V_{G1})$ and $G_{2}(E_{G2},V_{G2})$, with scalar weights on the vertices, I would like to find a subgraph $H_{1}$ of $G_{1}$ that best matches some subgraph $H_{2}$ of $G_{2}$.  However, in most papers I've seen the weights are usually on the edges, $H_{1}$ is pre-specified and the matching being attempted was exact. I would be grateful if somebody could point me in the right direction.Clarification edit: I'm looking for $H_1(E_{H1},V_{H1})\subset G1(E_{G1},V_{G1})$ and an injective function $f:V_{H1}\rightarrow V_{G2}$ so that $|\{(f(u),f(v))$ $\in$ $E_{G2}$ | $(u,v) \in E_{H1}\}| / |E_{H1}|$ is maximized while the sum of the node weight differences - $\sum_{v \in V_{H1}} |w(f(v))-w(v)|$ is minimized.     
What are the popular approaches to inexact attributed-subgraph matching?
algorithms;graphs