Shahnawaz Ahmad Rather

arXiv2026

Binomial edge ideals of bipartite complements of even cycles

Let $G_n$ be the bipartite complement of the even cycle $C_{2n}$, and let $J_{G_n}$ be its binomial edge ideal. For $n\geq5$, we study the interaction between the combinatorial structure of $G_n$ and the algebraic invariants of $S_{G_n}/J_{G_n}$. Our main combinatorial ingredient is a classification of the subsets of $V(G_n)$ having the cut point property, obtained through an analysis of disconnected induced subgraphs of $G_n$. We use this classification to describe the minimal primes and to study dimension and degree-theoretic invariants of $J_{G_n}$ and $S_{G_n}/J_{G_n}$. We also study local Vasconcelos numbers, graded Betti numbers and the Hilbert series, and obtain information on projective dimension and depth. Finally, we analyze induced paths in $G_n$ and derive corresponding bounds for the Castelnuovo--Mumford regularity.

math.AC↗

arXiv2026

Homological invariants of Edge Ideals associated to powers of cycles

Let $G_{n,m}=\overline{\C_n^{[m]}}$, where $\C_n^{[m]}$ denotes the closed $m$th power of the $n$-cycle. We study the graded Betti numbers and homological invariants of the edge ring of $G_{n,m}$ in the range $n\geq 3m+1$. These graphs form a natural family for the study of edge rings whose regularity can be compared explicitly with the induced matching number. In particular, for $n\geq4m+1$, the graph $G_{n,m}$ has induced matching number one, whereas its edge ring has regularity two. Our approach is based on a characterization of the homology of the induced subcomplexes of the independence complex $Δ(G_{n,m})$. We introduce a family $\mathcal{S}_V(k,m)$ of vertex subsets characterized by their successive gaps around the cycle and show that, for $W\in\mathcal{S}_V(k,m)$, the induced subcomplex $Δ[W]$ has the homotopy type of $\mathbb{S}^1$, whereas for $W\notin\mathcal{S}_V(k,m)$ all its positive-dimensional reduced homology groups vanish. Combining this characterization with Hochster's formula and an explicit enumeration of $\mathcal{S}_V(k,m)$, we obtain a closed formula for the graded Betti numbers in the second strand. We further determine the extremal Betti number, regularity, and projective dimension of the edge ring of $G_{n,m}$. Finally, we compute the $f$- and $h$-vectors of the independence complex and use the Hilbert series to determine the graded Betti numbers in the linear strand. The case $m=2$ recovers the corresponding results for complements of squares of cycles obtained in~\cite{RatherSquare}.

math.AC↗