Vertex decomposability and regularity of very well-covered graphs

Matchings, coverings, and Castelnuovo-Mumford regularity

“…In particularly we give a geometric proof of the well-known fact (Lemma 7) that reg (R/I(G)) is at least the induced matching number of G. In Section 4, we combine Theorem 1 with results from the graph theory literature to prove Theorems 2 and 3. We recover and extend results of [18] and [24], but show that results of [26] and [33] cannot be proved using this technique.…”

“…It is obvious that every well-covered bipartite graph is very well-covered. Mahmoudi et al [26] generalized Theorem 15 to show: Theorem 20. (Mahmoudi, Mousivand, Crupi, Rinaldo, Terai, and Yassemi [26]) If G is a very well-covered graph, then reg (R/I(G)) = indmatch G.…”

Algebraic Properties of Product of Graphs

“…Kummini [12] proved that A contains unmixed bipartite graphs. Also the authors in [13] generalized Kummini's result to the class of very well-covered graphs. In addition, A contains all cycles C m that m ≡ 0 and 1 mod 3 (see [10]).…”

“…Kummini [12] proved equality holds also for unmixed bipartite graphs. In addition, the authors in [13] generalized Kummini's result to the class of very well-covered graphs.…”

“…Our first topic in this paper is about the regularity of product. The regularity is one of the most important invariants of a graded module so that the regularity of edge rings has been a subject of study in combinatorial commutative algebra (see for example, [23,6,20,12,13]). The (Castelnuovo-Mumford) regularity of a graded R-module M, denoted by reg(M ), is defined by reg(M ) = max{j − i | β i,j (M ) = 0}.…”