We prove that for any unweighted graph on n vertices the L1 norm of a unit electric current between the endpoints of a random edge is at most 2 log n. Furthermore, we show that on any weighted graph the spectral norm of the entry-wise absolute value of the symmetric transfer-current matrix is at most 2 log n. This bound is tight up to constants and improves the O(log^2 n) bound from [Schild-Rao-Srivastava, SODA '18]. The initial proofs were generated by OpenAI's ChatGPT 5.5 Pro; the authors have verified and rewritten them to enhance readability and provide additional context.

翻译：我们证明：对于任意不含权的n顶点图，随机边两端点之间单位电流的L1范数不超过2 log n。进一步地，我们证明在任意赋权图上，对称转移电流矩阵元素绝对值之谱范数不超过2 log n。该界在常数意义下是紧的，且改进了[Schild-Rao-Srivastava, SODA '18]中的O(log^2 n)界。初始证明由OpenAI的ChatGPT 5.5 Pro生成；作者已对其进行验证并重写以增强可读性并提供更多背景信息。