Regularity bound of generalized binomial edge ideal of graphs

Let G be a simple graph on n vertices and JKm,G be the generalized binomial edge ideal of G in S=K[xi,j:i∈[m],j∈[n]]. We give a regularity upper bound of S/JKm,G. We prove that the regularity of S/JKm,G is bounded above by n−1. If m≥n, then we show that the regularity of S/JKm,G is n−1. We prove that if m<n, then the regularity of S/JKm,G is bounded below by max⁡{m−1,l(G)}, where l(G) is the length of longest induced path in G. Indeed, we prove Saeedi Madani-Kiani regularity upper bound conjecture for chordal graphs.