Consider a graph $G=(V,E)$ with positive edge weights. For simplicity, in my question, you can also assume unweighted graph. I am trying to show that the Laplacian quadratic form of the graph consisting of the $V$ nodes with only one edge $\{ u, v \} \in E$ in it is at most that of the whole graph $G$ times the effective resistance of edge $\{u,v\} \in G$, i.e.,
$$ x^T L_{u,v} x \leq R_{u,v}[G] x^T L_G x, \text{ for any vector } x \in \mathbb{R}^{n \times 1} $$
and this was claimed that it is known on page 6 of this paper, before the corollary. Any hint would be appreciated.