Imran Anwar

f-Ideals of Degree 2

In this paper, we introduce the concept of f-ideals and discuss its algebraic properties. In particular, we give the characterization of all the f-ideals of degree 2.

ON THE CHARACTERIZATION OF f-IDEALS

In this paper, we give the characterization of unmixed f-ideals of degree d ≥ 2 generalizing the results given in [1].

Quasi-linear Quotients and Shellability of Pure Simplicial Complexes

For a square-free monomial ideal I ⊂ S = k[x 1, x 2,…, x n ], we introduce the notion of quasi-linear quotients. By using the quasi-linear quotients, we give a new algebraic criterion for the shellability of a pure simplicial complex Δ over [n]. Also, we provide a new criterion for the Cohen–Macaulayness of the face ring of a pure simplicial complex Δ. Moreover, we show that the face ring of the spanning simplicial complex (defined in [2]) of an r-cyclic graph is Cohen–Macaulay.

Spanning Simplicial Complexes of Uni-Cyclic Graphs

In this paper, we introduce the concept of the spanning simplicial complex Δs(G) associated to a simple finite connected graph G. We characterize all spanning trees of the uni-cyclic graph Un,m. In particular, we give a formula for computing the Hilbert series and h-vector of the Stanley-Reisner ring k[Δs(Un,m)]. Finally, we prove that the spanning simplicial complex Δs(Un,m) is shifted and hence is shellable.

A construction of Cohen–Macaulay f-graphs

In this paper, we define and characterize the f-graphs. Also, we give a construction of f-graphs and importantly we show that the f-graphs obtained from this construction are Cohen–Macaulay.

On the connectedness of f-simplicial complexes

In this paper, we introduce the concept of f-simplicial complexes by generalizing the term of f-graphs (introduced in [H. Mahmood, I. Anwar and M. K. Zafar, Construction of Cohen–Macaualy f-graphs, J. Algebra Appl. 13(6) (2014) 1450012]). In particular, we discuss the problem of connectedness of pure f-simplicial complexes. Moreover, we give a complete characterization of connected and disconnected f-graphs and give a classification of all the disconnected f-graphs.

On algebraic characterization of SSC of the Jahangir’s graph n,m

Abstract In this paper, some algebraic and combinatorial characterizations of the spanning simplicial complex Δs(𝓙n,m) of the Jahangir’s graph 𝓙n,m are explored. We show that Δs(𝓙n,m) is pure, present the formula for f-vectors associated to it and hence deduce a recipe for computing the Hilbert series of the Face ring k[Δs(𝓙n,m)]. Finally, we show that the face ring of Δs(𝓙n,m) is Cohen-Macaulay and give some open scopes of the current work.

On Characteristic Poset and Stanley Decomposition of S=I

Let K be a field and S = K[x1,…,xn] be the polynomial ring in n variables. Let I ⊂ S be a monomial ideal such that S/I is Cohen-Macaulay. By associating a finite poset to S/I, we show that if S/I is a Stanley ideal then T/Ĩ is also a Stanley ideal, where T = K[x11,…,x1a1,…,xn1,…,xnan] and Ĩ is the polarization of I.

On Characteristic Poset and Stanley Decomposition

Abstract Let J ⊂ I be two monomial ideals such that I/J is Cohen Macaulay. By associating a finite posets PI/Jg

INCLUSION IDEALS ASSOCIATED TO UNIFORMLY INCREASING HYPERGRAPHS

P_{I/J}^g