Regularity of binomial edge ideals of certain block graphs

Abstract

We prove that the regularity of binomial edge ideals of graphs obtained by gluing two graphs at a free vertex is the sum of the regularity of individual graphs. As a consequence, we generalize certain results of Zafar and Zahid. We obtain an improved lower bound for the regularity of trees. Further, we characterize trees which attain the lower bound. We prove an upper bound for the regularity of certain subclass of blockgraphs. As a consequence we obtain sharp upper and lower bounds for a class of trees called lobsters.

Krull dimension and regularity of binomial edge ideals of block graphs

“…Recently, in [9] Herzog and the second author computed one of the distinguished extremal Betti number of the binomial edge ideal of a block graph and classify all block graphs admitting precisely one extremal Betti number giving a natural lower bound for the regularity of any block graph. Jayanthan et al in [10] and in [11] obtained a related result for trees, a subclass of block graphs.…”

Regularity of binomial edge ideals of Cohen-Macaulay bipartite graphs

“…Therefore, reg(S H /J H ) = r − 1. By[8, Theorem 3.1] and the claim, we get reg(S/J Gv 23 ) = r. Now we show that reg(S/Q v 23 ) ≤ r − 2 and reg(S/(Qv 23 + J Gv 23 )) ≤ r − 1. It follows from [14, Proposition 2.3] that if T ∈ C (G) and v 23 ∈ T , then v 21 ∈ T and v 24 / ∈ T .…”

“…If G is not decomposable, then it is called an indecomposable graph. It follows from [8,Theorem 3.1] that to find the regularity, it is enough to consider G to be an indecomposable graph. So, for the rest of the section, we assume that G is an indecomposable graph.…”

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

“…In Theorem 2.4 we classify all block graphs with the property that they admit precisely one extremal Betti number, by listing the forbidden induced subgraphs (which are 4 in total), and we also give an explicit description of the block graphs with precisely one extremal Betti number. Carla Mascia informed us that Jananthan et al in an yet unpublished paper and revised version of [10] obtained a related result for trees.…”