Graded Betti numbers of some circulant graphs

TL;DR

This paper calculates the graded Betti numbers of edge ideals for specific families of circulant graphs and explores their algebraic and combinatorial properties such as regularity, Cohen-Macaulayness, and well-coveredness.

Contribution

It provides explicit Betti number formulas for certain circulant graphs and analyzes their algebraic and combinatorial characteristics, extending understanding of their structural properties.

Findings

Computed Betti numbers for three families of circulant graphs.

Determined conditions for properties like Cohen-Macaulayness and Buchsbaum.

Analyzed regularity, projective dimension, and other invariants.

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.