[Paper Review] 2-dimensional vertex decomposable circulant graphs

Francesco Romeo, Giancarlo Rinaldo|arXiv (Cornell University)|Jul 16, 2018

Graph theory and applications15 references3 citations

TL;DR

This paper characterizes 2-dimensional vertex decomposable independence complexes of circulant graphs and constructs an infinite family of such graphs using specific generator sets. It proves that the independence complex of $ C_n(S) $ with $ n = 3 \cdot 2^m $, $ S = \{1,2,4,\dots,2^m, 2^m-1\} $, is vertex decomposable and shows that the corresponding Stanley-Reisner ring is a level algebra, with the only Gorenstein case being $ C_6(3) $.

ABSTRACT

Let $G$ be the circulant graph $C_n(S)$ with $S\subseteq\{ 1,\ldots,\left \lfloor\frac{n}{2} ight floor\}$ and let $Δ$ be its independence complex. We describe the well-covered circulant graphs with 2-dimensional $Δ$ and construct an infinite family of vertex-decomposable circulant graphs within this family.

Motivation & Objective

• To characterize well-covered circulant graphs whose independence complex is 2-dimensional and vertex decomposable.
• To construct an infinite family of vertex-decomposable circulant graphs within this class.
• To study the algebraic properties of the Stanley-Reisner ring, particularly whether it is a level or Gorenstein algebra.
• To determine necessary and sufficient conditions for the Stanley-Reisner ring of such graphs to be Gorenstein.
• To analyze the $ f $-vector, $ h $-vector, and reduced Euler characteristic of the independence complex for structural and algebraic insights.

Proposed method

• Uses the independence complex $ \Delta $ of a circulant graph $ G = C_n(S) $, defined as the simplicial complex of all independent sets in $ G $.
• Applies Hochster’s formula to compute Betti numbers of the edge ideal $ I(G) $, linking homology of links and deletions to Betti numbers.
• Employs the condition of vertex decomposability: for a vertex $ v $, both $ \operatorname{link}_\Delta(v) $ and $ \operatorname{del}_\Delta(v) $ must be vertex decomposable.
• Uses the $ f $-vector and $ h $-vector formulas (Propositions 2.2 and 2.4) to compute topological invariants like the reduced Euler characteristic $ \widetilde{\chi}(\Delta) $.
• Applies the Auslander-Buchsbaum formula to relate projective dimension and depth, and uses the regularity of $ R/I(G) $ to constrain Betti numbers.
• Uses the symmetry of the $ h $-vector as a necessary condition for Gorenstein property, and solves $ h_1 = h_2 $ to identify candidate graphs.

Experimental results

Research questions

• RQ1 Which circulant graphs have 2-dimensional independence complexes that are vertex decomposable?
• RQ2 Can an infinite family of vertex-decomposable circulant graphs with Krull dimension 3 be constructed?
• RQ3 When is the Stanley-Reisner ring of such a graph a level algebra?
• RQ4 Under what conditions is the Stanley-Reisner ring Gorenstein?
• RQ5 What is the role of the reduced Euler characteristic $ \widetilde{\chi}(\Delta) $ in determining the Cohen-Macaulay type?

Key findings

• The independence complex of $ C_n(S) $ with $ n = 3 \cdot 2^m $, $ S = \{1,2,4,\dots,2^m, 2^m-1\} $, is vertex decomposable for $ m \geq 3 $, forming an infinite family of such graphs.
• The Stanley-Reisner ring $ R/I(G) $ of this family is a level algebra, as the Cohen-Macaulay type equals the non-zero reduced Euler characteristic $ \widetilde{\chi}(\Delta) $.
• The only Gorenstein Stanley-Reisner ring in this class occurs for $ G = C_6(3) $, where $ \widetilde{\chi}(\Delta) = 1 $ and the $ h $-vector is symmetric.
• For 2-dimensional vertex decomposable independence complexes of circulants, the reduced Euler characteristic is always non-zero, ensuring the ring is level.
• The regularity of $ R/I(G) $ is 3, and the projective dimension is $ n - 3 $, derived from the Auslander-Buchsbaum formula.
• The $ f $-vector of $ \Delta(C_6(3)) $ is $ (1,6,12,8) $, yielding $ \widetilde{\chi}(\Delta) = -1 + 6 - 12 + 8 = 1 $, confirming the Gorenstein property.