Consider an $n$-node undirected graph $G = (V, E)$ equipped with weights $W$. Let $L$ be the weighted graph Laplacian matrix, i.e. $L_{ij} = -W_{(i,j)}$ for $(i,j)\in E$ and $L_{ii} = \sum_{j:(i,j)\in E} W_{(i,j)}$. Let $L^\dagger$ be the associated pseudo-inverse. I'm interested in controlling the follow quantity: let $s, t \in V$ $$F_{s, t} :=\sum_{(i, j)\in E} |(\mathbf{e}_i - \mathbf{e}_j)^\top L^\dagger (\mathbf{e}_s - \mathbf{e}_t)|$$ where $\mathbf{e}_i$ is simply the standard unit vector with 1 on $i$-th coordinate and 0 on the rest. I'm mostly interested in the case of an Erdős–Rényi graph $G\sim G(n, p)$. What's the tightest bound for $F_{s,t}$?
Any suggestion or reference would be appreciated!
