A Construction of Sequentially Cohen–Macaulay Graphs

For every simple graph [Formula: see text], a class of multiple clique cluster-whiskered graphs [Formula: see text] is introduced, and it is shown that all such graphs are vertex decomposable; thus, the independence simplicial complex [Formula: see text] is sequentially Cohen–Macaulay. The properties of the graphs [Formula: see text] and [Formula: see text] constructed by Cook and Nagel are studied, including the enumeration of facets of the complex [Formula: see text] and the calculation of Betti numbers of the cover ideal [Formula: see text]. We also prove that the complex[Formula: see text] is strongly shellable and pure for either a Boolean graph [Formula: see text] or the full clique-whiskered graph [Formula: see text] of [Formula: see text], which is obtained by adding a whisker to each vertex of [Formula: see text]. This implies that both the facet ideal [Formula: see text] and the cover ideal [Formula: see text] have linear quotients.

A note on f-graphs

The notion of [Formula: see text]-ideal was introduced in [G. Q. Abbasi, S. Ahmad, I. Anwar and W. A. Baig, [Formula: see text]-Ideals of degree 2, Algebra Colloq. 19(1) (2012) 921–926] in 2012 and has been studied in many papers after that. In this paper, we have studied those graphs whose Stanley–Reisner ideals turn out to be [Formula: see text]-ideals. We give a characterization and construction of these graphs and show that, unlike [Formula: see text]-graphs, these graphs are always connected. We have also discussed when these graphs are complete bipartite graphs. Moreover, we classify those graphs for which both the facet ideal (the edge ideal) and the Stanley–Resiner ideal, are [Formula: see text]-ideals.

Spanning Simplicial Complex of Wheel Graph Wn

In this paper, we explore the spanning simplicial complex of wheel graph Wn on vertex set [n]. Combinatorial properties of the spanning simplicial complex of wheel graph are discussed, which are then used to compute the f-vector and Hilbert series of face ring k[Δs(Wn)] for the spanning simplicial complex Δs(Wn). Moreover, the associated primes of the facet ideal [Formula: see text] are also computed.

Cohen–Macaulay modifications of the facet ideal of a simplicial complex

We define the chordal simplicial complex by using the definition of chordal clutter introduced by Woodroofe. We show that the facet ideal of the chordal simplicial complex is Cohen–Macaulay if and only if it is unmixed. Moreover, we prove that the facet ideal of a chordal simplicial complex has infinitely many nontrivial Cohen–Macaulay modifications.