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arxiv: 2606.22513 · v1 · pith:C7PBFBFRnew · submitted 2026-06-21 · 🧮 math.AC · math.CO

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

Mohammed Rafiq Namiq This is my paper

Pith reviewed 2026-06-26 09:21 UTC · model grok-4.3

classification 🧮 math.AC math.CO
keywords sequentially Cohen-Macaulay graphsco-chordal graphsBetti numbersedge idealsglued clique complexes(d1,...,dq)-treeszero-divisor graphssplit graphs
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The pith

Sequentially Cohen-Macaulay co-chordal graphs receive explicit graded Betti number formulas for their edge ideals once their chordal complements are identified as (d1,...,dq)-trees.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper establishes that the chordal complements of sequentially Cohen-Macaulay co-chordal graphs can be viewed as (d1,...,dq)-trees, which in turn supply explicit formulas for the graded Betti numbers of the associated edge ideals. This produces a complete homological characterization of the graphs in question. The same approach yields exact invariants and Cohen-Macaulay classifications for split graphs, threshold graphs, prime ideal graphs, nilpotent graphs over Artinian chain rings, and zero-divisor graphs of Z_n, where the latter are sequentially Cohen-Macaulay precisely when n equals 2p or p^a for prime p.

Core claim

Using the characterization of these complements as (d1,…,dq)-trees, we derive explicit formulas for the graded Betti numbers of the associated edge ideals, yielding a complete homological characterization of sequentially Cohen-Macaulay co-chordal graphs. The same formulas give exact invariants for split graphs, threshold graphs, and prime ideal graphs, classify their Cohen-Macaulay cases, characterize sequentially Cohen-Macaulay nilpotent graphs of finite direct products of Artinian chain rings with a closed formula in terms of local nilpotency indices and residue field cardinalities, and classify the zero-divisor graphs of Z_n as sequentially Cohen-Macaulay exactly when n=2p or n=p^a for pr

What carries the argument

The characterization of chordal complements as (d1,…,dq)-trees, which supports derivation of the graded Betti numbers via glued clique complexes of the edge ideals.

If this is right

  • Split graphs, threshold graphs, and prime ideal graphs receive exact homological invariants together with a classification of their Cohen-Macaulay cases.
  • Nilpotent graphs of finite direct products of Artinian chain rings are characterized as sequentially Cohen-Macaulay with a closed Betti formula in terms of nilpotency indices and residue field sizes.
  • Zero-divisor graphs of Z_n are sequentially Cohen-Macaulay if and only if n=2p or n=p^a for prime p.
  • The glued clique complex construction supplies the explicit Betti formulas once the tree characterization is in place.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The same tree-based formulas could be tested on additional families whose complements admit a (d1,...,dq)-tree description.
  • The classification for zero-divisor graphs of Z_n suggests checking whether the same n-condition governs sequential Cohen-Macaulayness in related algebraic graph constructions.
  • Explicit Betti numbers open the possibility of computing projective dimensions or regularity directly from the parameters d1,...,dq without resolving the ideal.

Load-bearing premise

The complements of the graphs under study can be characterized as (d1,…,dq)-trees.

What would settle it

A counterexample would be a sequentially Cohen-Macaulay co-chordal graph whose chordal complement is not a (d1,…,dq)-tree, or a mismatch between the derived Betti number formula and direct computation for any listed family such as the zero-divisor graph of Z_6.

Figures

Figures reproduced from arXiv: 2606.22513 by Mohammed Rafiq Namiq.

Figure 1
Figure 1. Figure 1: The split graph structure of G. 5.2. Complete, Nearly Complete, and Pineapple Graphs. The exact expression in Theorem 5.2 admits several important specializations corresponding to uniform or nearly uniform bipartite attachments between the clique and the independent set. Applying our unified formula, the following corollaries provide the exact homological values for the specific subclasses recently studied… view at source ↗
Figure 2
Figure 2. Figure 2: The zero-divisor graph Γ(Z27). 0 1 2 3 4 5 6 7 total: 1 13 42 70 70 42 14 2 0 1 . . . . . . . 1 . 13 42 70 70 42 14 2 In particular, pdim(S/I(Γ(Z27))) = 7, since β7 ̸= 0 while βi = 0 for all i > 7. Corollary 6.8. Let n ≥ 2. Then S/I(Γ(Zn)) is Cohen–Macaulay if and only if n = p or n = p 2 for some prime p. Proof. By Theorem 6.6, the graph Γ(Zn) is sequentially Cohen–Macaulay if and only if n = p a or n = 2… view at source ↗
read the original abstract

We study sequentially Cohen-Macaulay co-chordal graphs through the glued clique complexes of their chordal complements. Using the characterization of these complements as $(d_1,\ldots,d_q)$-trees, we derive explicit formulas for the graded Betti numbers of the associated edge ideals, yielding a complete homological characterization of sequentially Cohen-Macaulay co-chordal graphs. As applications, we determine exact homological invariants for several important graph families, including split graphs, threshold graphs, and prime ideal graphs, and classify their Cohen-Macaulay cases. We further characterize the sequentially Cohen-Macaulay nilpotent graphs of finite direct products of Artinian chain rings and establish a closed formula for their graded Betti numbers in terms of local nilpotency indices and residue field cardinalities. Finally, we classify the zero-divisor graphs of $\mathbb{Z}_n$, proving that they are sequentially Cohen-Macaulay if and only if $n=2p$ or $n=p^a$, where $p$ is a prime.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

0 major / 2 minor

Summary. The paper claims that sequentially Cohen-Macaulay co-chordal graphs can be studied via the glued clique complexes of their chordal complements. It asserts that these complements are characterized as (d1,…,dq)-trees, from which explicit formulas for the graded Betti numbers of the associated edge ideals are derived, giving a complete homological characterization. Applications include exact invariants and Cohen-Macaulay classifications for split graphs, threshold graphs, and prime ideal graphs; a characterization and closed Betti formula for sequentially Cohen-Macaulay nilpotent graphs of finite direct products of Artinian chain rings; and a classification showing that the zero-divisor graph of Zn is sequentially Cohen-Macaulay precisely when n=2p or n=p^a for prime p.

Significance. If the tree characterization and the resulting Betti formulas are valid, the work supplies explicit homological data for several standard graph families and resolves the sequentially Cohen-Macaulay property for zero-divisor graphs of Zn. Such formulas would be directly usable for further computations in algebraic combinatorics and could serve as test cases for broader conjectures on sequentially Cohen-Macaulay edge ideals.

minor comments (2)
  1. The abstract paragraph on glued clique complexes invokes the (d1,…,dq)-tree characterization without indicating where the justification appears; a forward reference to the relevant theorem or section would improve readability.
  2. The applications section lists several graph families; it would help to state explicitly which of the derived Betti formulas are new versus recovered from earlier literature.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their careful summary of the manuscript. No specific major comments or points of criticism were listed in the report, so we have no individual items to address point-by-point. We remain available to clarify any aspect of the tree characterization, Betti formulas, or applications if the referee or editor requests further details.

Circularity Check

0 steps flagged

No circularity; derivation self-contained against external characterization

full rationale

The abstract invokes an external characterization of complements as (d1,…,dq)-trees to derive Betti formulas, but supplies no equations or self-citations that reduce the claimed predictions back to fitted inputs or prior self-results by construction. The skeptic analysis confirms that without supplied full text exhibiting such a reduction, the central homological characterization rests on the stated premise rather than re-deriving it tautologically. This is the normal case of an independent derivation once the tree characterization is granted.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract-only review supplies no information on free parameters, background axioms, or new entities introduced by the paper.

pith-pipeline@v0.9.1-grok · 5716 in / 1100 out tokens · 29775 ms · 2026-06-26T09:21:30.497355+00:00 · methodology

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Reference graph

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