AI 中文总结
本文证明线图运算不增加电阻直径，即对任意有限连通简单图，其线图的电阻直径不超过原图，等号当且仅当图为圈或K4，方法结合星形电路表示与分支核心预算不等式。
AI 中文摘要
我们证明了Xu、Li、Hua和Pan提出的猜想：在线图运算下，电阻直径不会增大。对于每个至少含有一条边的有限连通简单图$G$，我们建立了$D_r(L(G))\le D_r(G)$，当且仅当$G$是圈或$K_4$时取等号。证明将$L(G)$通过星形图精确地表示为电路，并结合分支核心的锐利预算不等式。该不等式控制了由度二路径比较产生的组合误差项，对直径界和等号分析均至关重要。
英文摘要
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.
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