Let $G = (V, E)$ be an undirected, connected graph of $n$ vertices, ordered from 1 to $n$. I have two weight functions $w_1, w_2 : E \to \mathbb{R}$, where $w_1$ is just given by $w_1(e) = 1$, $\forall e \in E$, and $w_2$ is such that \begin{align} \sum_{(i, j) \in V \times V} w_2(i, j) \leq 2m, \end{align} where $m$ is the number of edges in $G$. Of course, $w_1(i, j) = w_2(i, j) = 0$ when $(i, j) \notin E$.
Let $C_{xy}$ be the expected commute time between $x$ and $y$ of a random walk, then by interpreting the graph $G$ as an electrical network it is a well known result that \begin{align} C_{xy} = R_{xy} \cdot \sum_{(i, j) \in V \times V} w(i, j), \end{align} where $R_{xy}$ is the effective resistance between vertices $x$ and $y$.
I wish to prove that $C^2_{1n} \leq C^1_{1n}$ on a particular graph, but I'm having problems bounding the increase in effective resistance between 1 and $n$ when moving from weights $w_1$ to weights $w_2$.
I know that most weights in $w_2$ decrease below 1, while only a handful increase above one, but I don't know how to use this to bound the effective resistance.
There doesn't seem to be much literature on this either, as the only result I've found is that the change in effective resistance on $ij$ when $w(a, b)$ is increased by $\alpha > 0$ is equal to \begin{align} \Delta R_{ij}(a, b, \alpha) = \frac{\alpha}{1 + \alpha L(b)^{-1}_{aa}} \left(L(b)^{-1}_{ai} - L(b)^{-1}_{aj} \right)^2, \end{align} where $L(b)$ is the laplacian of $G$ with row and column $b$ removed.
But this result is too complex to use, I think. This is probably a long shot, but if anyone has any ideas or can point me in the right direction, I would be very thankful!