We prove the upper bound conjecture proposed by Saeedi Madani and 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. © The Author(s), under exclusive licence to Universitat de Barcelona 2024.

This research focuses on analyzing the depth of generalized binomial edge ideals. We extend the notion of d-compatible map and use it to give a combinatorial lower bound for the depth of generalized binomial edge ideals. Subsequently, we determine an upper bound for the depth of generalized binomial edge ideals in terms of the vertex-connectivity of graphs. We demonstrate that the difference between the upper and lower bounds can be arbitrarily large, even in cases when one of the bounds is sharp. In addition, we calculate the depth of generalized binomial edge ideals of certain classes of graphs, including cycles and graphs with Cohen-Macaulay binomial edge ideals. © The Author(s), under exclusive licence to Springer Nature Switzerland AG 2024.

We show that the minimal number of generators and the Cohen-Macaulay type of a family of numerical semigroups generated by concatenation of arithmetic sequences is unbounded.

We introduce the notion of numerical semigroups generated by concatenation of arithmetic sequences and show that this class of numerical semigroups exhibit multiple interesting behaviors.