From matrices to bipartite graphs

Question

Assume $G(A,B)$ is a bipartite graph and assume $L(G)$ is the adjacency matrix of its line graph. define $$B=[3\text{I}+L(G)]^{-1}$$. Is it always the case that for each edge $e=(a,b)\in G$, we have:

$$B_{e,e}>\sum_{e'\in G, e'=(a,b'),e'\neq e}B_{e,e'}$$

(I have asked the same question in Linear Algebra form here: Simple to state yet tricky question

Answer 1

I answered the linear algebra question referenced in this problem. You can read the answer here.