On a regularity-conjecture of generalized binomial edge ideals

In this paper, we prove the upper bound conjecture proposed by Saeedi Madani & Kiani on the Castelnuovo-Mumford regularity of generalized binomial edge ideals. We give a combinatorial upper bound of regularity for generalized binomial edge ideals, which is better than the bound claimed in that conjecture. Also, we show that the bound is tight by providing an infinite class of graphs.