On the binomial edge ideals of block graphs

Abstract

We find a class of block graphs whose binomial edge ideals have minimal regularity. As a consequence, we characterize the trees whose binomial edge ideals have minimal regularity. Also, we show that the binomial edge ideal of a block graph has the same depth as its initial ideal.

Krull dimension and regularity of binomial edge ideals of block graphs

“…T is a cutset of G if c(T \ {i}) < c(T ) for each i ∈ T , and we denote by C(G) the set of all cutsets for G. In [8] and [15] the authors gave a description of the primary decomposition of J G in terms of prime ideals induced by the set C(G) (see (2)). Thanks to this result the following formula for the Krull dimension is obtained (1) dim S/J G = max T ∈C(G)…”

On the Extremal Betti Numbers of Binomial Edge Ideals of Block Graphs

“…Now we prove the assertions regarding in < (J G ). If i(G) = 0, then J G is the ideal of 2-minors of the matrix (3). It is known that in < (J G ) has a 2-linear resolution.…”

“…Let J G denote the binomial edge ideal of a graph G. The first result showing that reg(J G ) = reg(in < (J G )) without computing all graded Betti numbers was obtained for PI graphs by Ene and Zarojanu [8]. Later Chaudhry, Dokuyucu and Irfan [3] showed that proj dim(J G ) = proj dim(in < (J G )) for any block graph G, and reg(J G ) = reg(in < (J G )) for a special class of block graphs. Roughly speaking, block graphs are trees whose edges are replaced by cliques.…”

Cohen-Macaulay generalized binomial edge ideals

“…One can observe that if (G 1 , G 2 ) = (K m , G), where K m is the complete graph on m vertices, J Km,G is the generalized binomial edge ideal associate to G. In [10] and [7], unmixedness and Cohen-Macaulayness of binomial edge ideal of a pair of graphs are characterized in some special cases. The algebraic properties of the class of generalized binomial edge ideals are widely open, although some results are known (see for instance [7,15,8] and the reference therein). An interesting problem is to determine a classification of the Cohen-Macaulay generalized binomial edge ideals.…”