Graded Betti numbers of some circulant graphs

TL;DR

This work addresses the computation of $\mathbb{N}$-graded Betti numbers for edge ideals of three structured families of circulant graphs, by expressing these graphs as joins of cycles or multipartite graphs and applying Hochster's formula alongside join-transport results. The authors provide explicit Betti-number formulas, determine regularity and projective dimension, and characterize induced matching numbers across the families. They also analyze a range of combinatorial-algebraic properties (well-covered, Cohen–Macaulay, sequentially Cohen–Macaulay, Buchsbaum, $S_2$) through the lens of the graph decomposition. The results illuminate how regularity and other invariants behave under graph joins and multipartite structures, contributing concrete, computable invariants for a broad class of circulant graphs.

Abstract

Let $G$ be the circulant graph $C_n(S)$ with $S \subseteq \{1, 2, \dots, \lfloor \frac{n}{2} \rfloor\}$, and let $I(G)$ denote the edge ideal in the polynomial ring $R=\mathbb{K}[x_0, x_1, \dots, x_{n-1}]$ over a field $\mathbb{K}$. In this paper, we compute the $\mathbb{N}$-graded Betti numbers of the edge ideals of three families of circulant graphs $C_n(1,2,\dots,\widehat{j},\dots,\lfloor \frac{n}{2} \rfloor)$, $C_{lm}(1,2,\dots,\widehat{2l},\dots, \widehat{3l},\dots,\lfloor \frac{lm}{2} \rfloor)$ and $C_{lm}(1,2,\dots,\widehat{l},\dots,\widehat{2l},\dots, \widehat{3l},\dots,\lfloor \frac{lm}{2} \rfloor)$. Other algebraic and combinatorial properties like regularity, projective dimension, induced matching number and when such graphs are well-covered, Cohen-Macaulay, Sequentially Cohen-Macaulay, Buchsbaum and $S_2$ are also discussed.

Figures (3)

• Figure 1: $C_{12}(1,\widehat{2},3,4,5,6) \cong C_6^c * C_6^c$
• Figure 2: $C_{18}(1,2,3,4,5,\widehat{6},7,8,\widehat{9}) \cong C_6 * C_6 *C_6$
• Figure 3: $C_{12}(1,\widehat{2},3,\widehat{4},5,\widehat{6}) \cong K_{6,6}$

Theorems & Definitions (47)

• Theorem 2.1
• Theorem 2.2
• Theorem 2.3
• Proposition 2.4
• Proposition 2.5
• Proposition 2.6
• Proposition 2.7
• proof
• Proposition 2.8
• proof