Regularity, Depth and Arithmetic Rank of Bipartite Edge Ideals

Regularity of second power of edge ideals

Let G be a graph with edge ideal I(G). Benerjee and Nevo proved that for every graph G, the inequality reg(I(G)2)≤reg(I(G))+2 holds. We provide an alternative proof for this result.

An upper bound for the regularity of powers of edge ideals

‎A recent result due to Ha and Van Tuyl proved that the Castelnuovo-Mumford regularity of the quotient ring $R/I(G)$ is at most matching number of $G$‎, ‎denoted by match$(G)$‎. ‎In this paper‎, ‎we provide a generalization of this result for powers of edge ideals‎. ‎More precisely‎, ‎we show that for every graph $G$ and every $sgeq 1$‎, ‎$${rm reg}( R‎/ ‎I(G)^{s})leq (2s-1) |E(G)|^{s-1} {rm ma...

Bounds for the Regularity of Edge Ideal of Vertex Decomposable and Shellable Graphs

On the binomial edge ideals of block graphs

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.

8 - Szemerédi ’ s Regularity Lemma

Szemerédi’s Regularity Lemma [18] tells us that every graph can be partitioned into a constant number of sets of vertices in such a way that for most of the pairs of sets in the partition, the bipartite graph of edges between them has many of the properties that one would expect in a random bipartite graph with the same expected edge density. Szemerédi originally used his lemma to prove his cel...