arXiv subjects

Mohammed Rafiq Namiq

Publications and source records attributed to Mohammed Rafiq Namiq.

10 recordsLinked to original sources

arXiv2026

Vertex Dismissibility and Scalability of Simplicial Complexe

We study three nonpure extensions of vertex decomposability, shellability, and Cohen-Macaulayness at the minimum facet dimension. For $b=\operatorname{indim}Δ$, strong vertex dismissibility gives a deletion-link recursion on the complex itself in which deletion does not decrease $b$, while vertex dismissibility and scalability require the pure initial skeleton $Δ^{[b]}$ to be vertex decomposable and shellable, respectively. We prove that a join is strongly vertex dismissible exactly when both factors are strongly vertex dismissible. We also show that strong vertex dismissibility implies that every skeleton $Δ^k=Δ^{[k]}$, $k\le b$, is vertex decomposable. When $\operatorname{indim}Δ\leq1$, weak connectedness is equivalent to vertex dismissibility, scalability, and initially Cohen-Macaulayness over some, equivalently every, field. For quasi-forests, these conditions are also equivalent to strong vertex dismissibility. Our main graph result classifies independence complexes of pseudoforests by showing that strong vertex dismissibility, vertex dismissibility, scalability, and initially Cohen-Macaulayness over some or every field are equivalent and are characterized by a componentwise independent domination criterion. Finally, we connect the combinatorial properties of the pure initial skeleton with algebraic properties of its Alexander dual, focusing on vertex splittability, linear quotients, and linear resolutions in fixed degree.

math.AC↗

arXiv2026

Explicit Betti Numbers for Skeletons of Chordal Clique Complexes and Their Alexander Duals

Let $Δ_{\mathbf r}=Δ_{\mathbf r}(n_1,\ldots,n_e)$ be the clique complex of a chordal graph with maximal cliques $S_1,\ldots,S_e$ in a leaf order, where $n_m=|S_m|$, $r_m=|S_{m+1}\cap(S_1\cup\cdots\cup S_m)|$, $\mathbf r=(r_1,\ldots,r_{e-1})$, and $N_{\mathbf r}$ is the number of vertices. We determine the graded Betti numbers $β_{i,j}\bigl(\mathbb K[(Δ_{\mathbf r})_{(k)}]\bigr)$ of every skeleton and derive explicit formulas for its regularity, projective dimension, depth, multiplicity, Cohen-Macaulayness, and initially Cohen-Macaulayness. We also compute its extremal Betti numbers and describe its integral homology and homotopy type. For the Alexander dual $Δ_{\mathbf r}^{\vee}$, we construct an explicit minimal multigraded resolution whose shifts are determined by $N_{\mathbf r}-n_m$ and $N_{\mathbf r}-r_m$, and obtain its graded and multigraded Betti numbers, canonical module, Cohen-Macaulay type, $a$-invariant, and Gorenstein criterion. We further show that every proper skeleton $(Δ_{\mathbf r}^{\vee})_{(k)}$ is Cohen-Macaulay and level, determine its Betti numbers and type, and characterize its Gorenstein cases. Finally, we show that the graded Betti table of $\mathbb K[Δ_{\mathbf r}^{\vee}]$ determines the multisets ${n_m}$ and ${r_m}$ up to a common shift.

math.AC↗

arXiv2026

Regularity of Symbolic Powers of Co-Chordal Edge Ideals

Let $G$ be a finite simple co-chordal graph with at least one edge, and let $I(G)$ be its edge ideal. Over an arbitrary field, we prove that the symbolic powers of $I(G)$ satisfy $\operatorname{reg} I(G)^{(s)}=2s$ for every $s\ge1$. Thus every symbolic power of $I(G)$ has a degree resolution. Using Takayama's formula and clique trees, we reduce the regularity problem to a topological one. Finite convex geometry then shows that nontrivial homology implies a suitable set of leaves, and a weighted counting argument gives the required regularity formula. Finally, we show that symbolic powers of co-chordal edge ideals are not necessarily componentwise linear.

math.AC↗

arXiv2026

A Complete Classification of 2-Linear Neighborhood Complexes

Let $G$ be a nonempty finite simple graph. We study when the Stanley-Reisner ideal of its neighborhood complex has a $2$-linear resolution. Combining Fröberg's theorem with the classical hypertree criterion, we obtain the following equivalent description in graph terms: $G$ is bipartite, its indexed open neighborhoods are Helly, and every induced cycle of length at least eight has a filling from each color class. This class properly contains the chordal bipartite graphs without isolated vertices. Hochster's formula gives all squarefree multigraded Betti numbers, while face counts determine the complete graded Betti table. If $G$ has $n$ vertices and $c$ connected components, then its Stanley-Reisner ring has terminal Betti number $2c-1$, projective dimension $n-1$, and depth one. We also determine the multiplicity and the initially Cohen-Macaulay and Cohen-Macaulay cases. A second formula separates degree data from overlaps caused by repeated common neighbors and yields closed expressions for bipartite graphs without $K_{2,3}$, cactus graphs, pseudoforests, and forests. For square cactus graphs, the Betti table recovers every degree multiplicity at least three; for forests, it recovers the complete degree sequence. Finally, the dominance complex has a $2$-linear Stanley-Reisner ideal precisely for nontrivial stars.

math.CO↗

arXiv2026

Betti Numbers of Sequentially Cohen-Macaulay Co-Chordal Graphs and Their Applications

We study edge ideals of sequentially Cohen-Macaulay co-chordal graphs through the maximal-clique structure of their chordal complements. After making the required construction order explicit, we derive a closed formula for the graded Betti numbers of a co-chordal graph whose complement is a $(d_1,\ldots,d_e)$-tree, and we characterize the Cohen-Macaulay case by its Betti sequence. The general formula is then specialized to split and threshold graphs, prime ideal graphs of finite rings, nilpotent graphs of products of finite chain rings, and zero-divisor graphs of finite chain rings. For products of chain rings, we characterize when the nilpotent graph is threshold and compute its Betti numbers in that range. Finally, among co-chordal zero-divisor graphs of $\mathbb{Z}_n$, we determine exactly when the quotient is sequentially Cohen-Macaulay: this occurs for $n=p^a$, $n=2p$, and $n=2p^2$, with $p$ odd in the last two cases.

math.AC↗

arXiv2025

The Graded Betti Numbers of the Skeletons of Simplicial Complexes

In this paper, we study a class $\mathcal{C}$ of squarefree monomial ideals $I\subseteq R=\mathbb{K}[x_1,\dots,x_n]$ over a field $\mathbb{K}$, defined by the condition that $\dim R/I$ equals the maximum degree of the minimal generators of $I$ minus one. We show that the Stanley-Reisner ideal of every $i$-skeleton of a simplicial complex $Δ$ belongs to $\mathcal{C}$ for all $-1\le i<\dimΔ$. To investigate their homological properties, we introduce the notion of a degree resolution and prove that every ideal in $\mathcal{C}$ possesses this property. Moreover, we show that every squarefree monomial ideal admits a truncation whose regularity coincides with that of the original ideal, thereby reducing the study of degree resolutions to that of linear resolutions. Finally, we provide an explicit formula describing the relationship between the graded Betti numbers of a simplicial complex and those of its skeletons.

math.AC↗

arXiv2025

Initially Cohen-Macaulay Modules

In this paper, we introduce initially Cohen-Macaulay modules over a commutative Noetherian local ring $R$, a new class of $R$-modules that generalizes both Cohen-Macaulay and sequentially Cohen-Macaulay modules. A finitely generated $R$-module $N$ is initially Cohen-Macaulay if its depth is equal to its initial dimension, an invariant defined as the infimum of the coheights of the associated primes of $N$. We develop the theory of these modules, providing homological, combinatorial, and topological characterizations and confirming their compatibility with regular sequences, localization, and dimension filtrations. When this theory is applied to simplicial complexes, we establish analogues of Reisner's criterion, the Eagon-Reiner theorem, and Duval's characterization of sequentially Cohen-Macaulay complexes. Finally, we classify certain classes of initially Cohen-Macaulay graphs of interest and those whose projective dimension coincides with their maximum vertex degree.

math.AC↗

arXiv2025

Sequentially Cohen-Macaulay Co-Chordal Graphs: Structure and Projective Dimension

We introduce a class of chordal graphs called ($d_1$,$d_2$,$\dots$,$d_q$)-trees. A graph belongs to this class if and only if its clique complex is sequentially Cohen-Macaulay, providing a complete classification of all sequentially Cohen-Macaulay co-chordal graphs. This class also yields a classification of bi-sequentially Cohen-Macaulay graphs. We study the relationship between the projective dimension of a graph and its maximum vertex degree. We show that the projective dimension is always at least the maximum vertex degree, although this bound is not always tight, even for co-chordal graphs. However, equality holds when the graph is sequentially Cohen-Macaulay co-chordal or has a full vertex.

math.AC↗

arXiv2022

The graded Betti numbers of truncation of ideals in polynomial rings

Let $R=\mathbb{K}[x_1,\dots,x_n]$, a graded algebra $S=R/I$ satisfies $N_{k,p}$ if $I$ is generated in degree $k$, and the graded minimal resolution is linear the first $p$ steps, and the $k$-index of $S$ is the largest $p$ such that $S$ satisfies $N_{k,p}$. Eisenbud and Goto have shown that for any graded ring $R/I$, then $R/I_{\geq k}$, where $I_{\geq k}=I\cap M^k$ and $M=(x_1,\dots,x_n)$, has a $k$-linear resolution (satisfies $N_{k,p}$ for all $p$) if $k\gg0$. For a squarefree monomial ideal $I$, we are here interested in the ideal $I_k$ which is the squarefree part of $I_{\geq k}$. The ideal $I$ is, via Stanley-Reisner correspondence, associated to a simplicial complex $Δ_I$. In this case, all Betti numbers of $R/I_k$ for $k>\min\{\text{deg}(u)\mid u\in I\}$, which of course is a much finer invariant than the index, can be determined from the Betti diagram of $R/I$ and the $f$-vector of $Δ_I$. We compare our results with the corresponding statements for $I_{\ge k}$. (Here $I$ is an arbitrary graded ideal.) In this case we show that the Betti numbers of $R/I_{\ge k}$ can be determined from the Betti numbers of $R/I$ and the Hilbert series of $R/I_{\ge k}$.

math.AC↗

arXiv2022

Quasi-forest simplicial complexes and almost Cohen-Macaulay

In this paper we study the quasi-forest simplicial complexes and we define the concept of simplicial $k$-cycle (denoted by $\mathcal{S}_k$) and simplicial $k$-point (denoted by $\mathcal{P}_k$). We show that a simplicial complex $Δ$ is quasi-forest if and only if it does not have any $\mathcal{P}_k$ and any $\mathcal{S}_k$ for $k\geq 3$. Furthermore we characterize almost Cohen-Macaulay quasi-forest simplicial complexes. In the end we show that the cycle graph $G=C_n$ is almost Cohen-Macaulay if and only if $n=3,4,5,6,7,8,9,11$.

math.AC↗