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Generalized binomial edge ideals of whisker graphs via an extension of generalized corona products

J Anuvinda, Ranjana Mehta, Kamalesh Saha

TL;DR

This work extends the study of generalized binomial edge ideals to whisker graphs via an expanded corona-product framework, introducing two graph classes $\mathcal{G}_1$ and $\mathcal{G}_2$ and examining depth, regularity, and Cohen–Macaulay properties. It proves a sharp depth lower bound for $\mathcal{G}_2$ and an exact depth formula for a broad subclass $\mathcal{G}'$ (encompassing whiskers), yielding $\mathrm{depth}(R/J_{K_m,D})=|V(D)|+(m-1)c(D)$, and in particular $\mathrm{depth}(R/J_{W(G)})=2|V(G)|+c(G)$. A tight regularity bound is established via the induced-matching number, with a precise calculation $\mathrm{reg}(R/J_{W(G)})=|V(G)|+1$ for whisker graphs on gap-free base graphs. The paper also provides a complete Cohen–Macaulay classification within $\mathcal{G}_2$, giving a new construction of CM binomial edge ideals and connecting CM-ness to the completeness of the base graph and CM-ness of the attached components. These results advance the structural understanding of generalized binomial edge ideals in broader graph families and offer concrete tools for computing depth, reg, and CM properties in this setting.

Abstract

In this paper, we initiate a systematic study of generalized binomial edge ideals of whisker graphs by working within a substantially broader class of graphs. We extend the notion of generalized corona products, and through this enlarged framework, investigate fundamental algebraic invariants such as depth, (Castelnuovo-Mumford) regularity, and the Cohen-Macaulay property. In particular, we establish a sharp lower bound on the depth of generalized binomial edge ideals for our extended class, and further obtain explicit depth formula for a broad subclass of this family, which in turn recovers the depth formula for whisker graphs. We also establish sharp upper bounds for the regularity, and in the case of binomial edge ideals of whisker graphs over gap-free graphs, determine the exact value of the regularity. Finally, for our extended class, we provide a combinatorial classification of all Cohen-Macaulay binomial edge ideals, which in turn yields a new construction of Cohen-Macaulay binomial edge ideals.

Generalized binomial edge ideals of whisker graphs via an extension of generalized corona products

TL;DR

This work extends the study of generalized binomial edge ideals to whisker graphs via an expanded corona-product framework, introducing two graph classes and and examining depth, regularity, and Cohen–Macaulay properties. It proves a sharp depth lower bound for and an exact depth formula for a broad subclass (encompassing whiskers), yielding , and in particular . A tight regularity bound is established via the induced-matching number, with a precise calculation for whisker graphs on gap-free base graphs. The paper also provides a complete Cohen–Macaulay classification within , giving a new construction of CM binomial edge ideals and connecting CM-ness to the completeness of the base graph and CM-ness of the attached components. These results advance the structural understanding of generalized binomial edge ideals in broader graph families and offer concrete tools for computing depth, reg, and CM properties in this setting.

Abstract

In this paper, we initiate a systematic study of generalized binomial edge ideals of whisker graphs by working within a substantially broader class of graphs. We extend the notion of generalized corona products, and through this enlarged framework, investigate fundamental algebraic invariants such as depth, (Castelnuovo-Mumford) regularity, and the Cohen-Macaulay property. In particular, we establish a sharp lower bound on the depth of generalized binomial edge ideals for our extended class, and further obtain explicit depth formula for a broad subclass of this family, which in turn recovers the depth formula for whisker graphs. We also establish sharp upper bounds for the regularity, and in the case of binomial edge ideals of whisker graphs over gap-free graphs, determine the exact value of the regularity. Finally, for our extended class, we provide a combinatorial classification of all Cohen-Macaulay binomial edge ideals, which in turn yields a new construction of Cohen-Macaulay binomial edge ideals.
Paper Structure (5 sections, 27 theorems, 61 equations, 2 figures)

This paper contains 5 sections, 27 theorems, 61 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Depth of generalized binomial edge ideals for $\mathcal{G}_2$
  4. Regularity of generalized binomial edge ideals for $\mathcal{G}_1$
  5. Cohen-Macaulay binomial edge ideals of graphs in $\mathcal{G}_2$

Key Result

Theorem \ref{thm:depth-bound-gen}

Let $D =G\circ_{S}(H_1,\ldots,H_{\ell})$ be a graph in $\mathcal{G}_{2}$. Then where $\ell$ is the cardinality of $S$, $p$ is the number of vertices in $G$, $c(D)$ is the number of connected components of $D$, $f(H_{i})$ denotes the number of free vertices in $H_{i}$ and $d(H_{i})$ denotes the sum of diameters of connected components of $H_{i}$.

Figures (2)

  • Figure 1: A graph in $\mathcal{G}_{1}$
  • Figure 2: A graph in $\mathcal{G}_{2}$

Theorems & Definitions (44)

  • Theorem \ref{thm:depth-bound-gen}
  • Theorem \ref{thm:depth-equal}
  • Theorem \ref{thm:depth-bound-binom}
  • Theorem \ref{thm:reg-bound-gen}
  • Theorem \ref{thm:reg-bound-gapfree}
  • Theorem \ref{thrm:cohen-macly}
  • Theorem 2.1: kumar20
  • Lemma 2.2: kumar20
  • Remark 2.3
  • Theorem 2.4: ams24
  • ...and 34 more