The first Steklov eigenvalue of graphs of positive genus

Let be a graph of genus with boundary . For , Lin and Zhao [J. Lond. Math. Soc. 112 (2025), Paper No. e70238] proved an upper bound for the first (non‐trivial) Steklov eigenvalue of , and they posed the problem of determining a corresponding bound for graphs of genus . In this paper, we prove an bound for a bounded‐degree graph of positive genus . Our result can be regarded as a discrete analogue of Kokarev's bound [Adv. Math. 258 (2014), 191–239], up to a constant factor.