ON THE HILBERT SERIES OF BINOMIAL EDGE IDEALS OF GENERALIZED TREES

Let G be a simple graph on the vertex set [n] = f1; : : : ; ng and K be a eld. Denition of binomial edge ideal rst appears independently in [3, 5]. The binomial edge ideal associated to G by denition is the ideal JG, generated by ffe : e = fi; j g2 E(G) and i < jg in R = RG = K[x1; : : : ; xn; y1; : : : ; yn], where fe = xiyj 􀀀 xjyi. In[1, 10, 5, 3], some algebraic properties of JG were studied and proved that JG is a radical ideal and determined when JG is a prime ideal. Recently Zafar has given a characterization of approximately Cohen-Macaulay binomial edge ideals for trees and proved that the binomial edge ideal of any cycle is approximately Cohen-Macaulay; in addition he computed the Hilbert series of the corresponding ideals[10]. The notion of closed graphs was introduced in [3] and Cohen-Macaulay closed graphs are completely classied in [1] and the Hilbert series of their binomial edge ideal is computed in[1]. Mohammadi and Sharifan used closed graphs with Cohen-Macaulay binomial edge ideals to compute the depth and the Hilbert function of further graphs. They computed the Hilbert function of a quasi cycle and gave a combinatorial description for the quotient ideal JG : fe and showed that JG : fe is a binomial edge ideal of another known graph in some cases[4]. In this paper we introduce the concept of generalized trees and generalized sun-graphs and compute the Hilbert series of binomial edge ideal of these classes of graphs.