arXiv subjects

Bilal Ahmad Wani

Publications and source records attributed to Bilal Ahmad Wani.

3 recordsLinked to original sources

arXiv2026

Edge ideals of complements of a class of circular interval graphs

Let $C_n^m$ be the $m$-th power of the cycle on $n$ vertices, and let $\Gnm$ be its complement, where $m\ge1$ and $n\ge3m+1$. We prove that every induced subcomplex of the clique complex of $C_n^m$ is homotopy equivalent either to a disjoint union of contractible complexes or to a circle, and we establish this dichotomy for a larger class of circular interval graphs. As a consequence, we obtain a closed formula for the linear strand of the Betti table of $R/I(\Gnm)$. We also prove that $R/I(\Gnm)$ is a Buchsbaum ring of depth $2$ and that the independence complex of $\Gnm$ is a triangulated $m$-manifold with boundary for $m\ge2$, namely a Möbius band or an annulus when $m=2$. For the complement $\overline H$ of every graph $H$ in the above class, we show that $\reg(I(\overline H)^k)=2k+\indm(\overline H)-1$ for all $k\ge2$, where $\indm$ denotes the induced matching number. In particular, some power of $I(\Gnm)$ has a linear resolution if and only if $n\ge4m+1$. In this range, we further prove that the colon ideals $(I^{k+1}:M)$, where $I=I(\Gnm)$ and $M$ is a minimal monomial generator of $I^k$, have regularity $2$.

math.AC↗

arXiv2026

Squarefree Powers of Edge Ideals under Graph Joins and Cones

For $q\ge 1$, the $q$-th squarefree power $I(G)^{[q]}$ of the edge ideal of a graph $G$ is generated by the squarefree monomials supported on $q$-matchings of $G$; it is the Stanley--Reisner ideal of the complex $Δ_q(G)=\{F\subseteq V(G):ν(G[F])<q\}$, where $ν$ denotes matching number. We prove a general formula for the matching number of an arbitrary graph join, \[ ν(G\ast H) = \min\Big(ν(G)+|V(H)|,\ \ ν(H)+|V(G)|,\ \ \Big\lfloor\tfrac{|V(G)|+|V(H)|}{2}\Big\rfloor\Big), \] via the Tutte--Berge formula, and use it to decompose $Δ_q(G\ast H)$ for arbitrary graphs $G,H$. Specializing to the wheel graph $\mathcal{W}_n = \mathcal{C}_n\ast\mathcal K_1$, we determine the Krull dimension and height of $R/I(\mathcal{W}_n)^{[q]}$ exactly for all $n\ge 3$, $1\le q\le\lfloor n/2\rfloor$, and -- combining our matching-number computations with a recent Tutte-type Cohen-Macaulayness criterion of Ficarra and Moradi -- prove that at the \emph{top} squarefree power $q=ν(\mathcal{W}_n)=\lceil n/2\rceil$, the ideal $I(\mathcal{W}_n)^{[ν(\mathcal{W}_n)]}$ is literally the squarefree Veronese ideal, so that $R/I(\mathcal{W}_n)^{[ν(\mathcal{W}_n)]}$ is Cohen-Macaulay with \[ {\rm dim} = {\rm depth} = {\rm reg}\big(R/I(\mathcal{W}_n)^{[ν(\mathcal{W}_n)]}\big) = 2\Big\lceil\frac n2\Big\rceil-1. \] This resolves all four classical invariants at the top power, and confirms there the pattern depth$(R/I(\mathcal{W}_n)^{[q]}) = 2q-1$ that our computational data (now extended to $n\le13$, every valid $q$) suggests holds throughout. We prove a general depth formula for squarefree powers of cone graphs, via a Betti-splitting exact sequence, that reduces this pattern to two more tractable statements about the underlying cycle alone; both are verified computationally in every case checked but left open in general.

math.AC↗

arXiv2026

The Weak Zero-Divisor Difference Graph of a Finite Commutative Ring

For a finite commutative ring $R$, let $\GR$ denote its zero-divisor graph and $\WGR$ its weakly zero-divisor graph, the latter containing the former as a spanning subgraph. We introduce the \emph{weak zero-divisor difference graph} $\DR:=\WGR-\GR$ and develop a complete structural theory for finite reduced rings $R\cong\mathbb F_{q_1}\times\cdots\times\mathbb F_{q_t}$. We show $\DR$ sits strictly between $\GR$ and $\WGR$ in a three-stage refinement that also contains Badawi's annihilator graph, and prove that distinct support classes $X_A,X_B$ are completely joined in $\DR$ if and only if $A\cap B\ne\emptyset$ -- an exact criterion underlying every result that follows. Consequently $\DR$ is edgeless for $t\le2$ but connected with diameter $2$ and girth $3$ for every $t\ge3$, independently of the field orders. We establish a complete perfectness dichotomy -- $\DR$ is perfect exactly when $t\in\{3,4\}$, with an elementary combinatorial proof at $t=4$, and never perfect for $t\ge5$ -- and determine its clique number exactly at $t=3$ and $t=4$; for general $t$ we give two incomparable lower bounds and two upper bounds, sharp at $t=3$ but not beyond, together with a compression argument showing an extremal family may always be taken shifted without this alone resolving the problem. A reconstruction theorem shows $\DR$ recovers the multiset of field orders intrinsically, so $\DR\cong\mathcal D(S)$ forces $R\cong S$. We further give closed forms, valid for every $t\ge3$, for the degree sequence and minimum degree, the domination number, and the independence and vertex cover numbers. Finally, we briefly indicate, via the valuation structure of finite chain rings, why the reduced-ring hypothesis cannot simply be dropped.

math.AC↗