A proof of the resistance diameter conjecture for line graphs

Source abstract

We prove the conjecture of Xu, Li, Hua, and Pan that the resistance diameter does not increase under the line-graph operation. For every finite connected simple graph $G$ with at least one edge, we establish $D_r(L(G))\le D_r(G)$, with equality if and only if $G$ is a cycle or $K_4$. The proof combines an exact electrical representation of $L(G)$ by stars with a sharp budget inequality for the branch core. This inequality controls the combined error terms arising from the comparison of degree-two paths and is central to both the diameter bound and the equality analysis.