Cinkir, Zubeyir2021-08-262021-08-2620160218-00060219-3094https://doi.org/10.1007/s00026-016-0304-2https://hdl.handle.net/20.500.12573/963This work is supported by The Scientific and Technological Research Council of Turkey-TUBITAK (Project No: 110T686) and by BAGEP of The Science Academy. I would like to thank the anonymous referees for their valuable suggestions, which improved this article in several ways. I also would like to thank Arzu Boysal and Fatih Ecevit for the helpful feedback on the earlier version of this article.Baker and Rumely's tau lower bound conjecture claims that if the tau constant of a metrized graph is divided by its total length, this ratio must be bounded below by a positive constant for all metrized graphs. We construct several families of metrized graphs having small tau constants. In addition to numerical computations, we prove that the tau constants of the metrized graphs in one of these families, the hexagonal nets around a torus, asymptotically approach to 108 which is our conjectural lower bound.enginfo:eu-repo/semantics/openAccesstau lower bound conjecturehexagonal net around a torustau constantmetrized grapharticleVolume 20 Issue 2 Page 317-344