A proof of the resistance diameter conjecture for line graphs

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.