Graded Betti numbers of some families 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 $C_n(1,2,\dots ,\widehat{\mathit{j}},\dots ,\lfloor \frac{n}{2}\rfloor )$, $C_{lm}(1,2,\dots ,\widehat{2l},\dots ,\widehat{3l},\dots ,\lfloor \frac{lm}{2}\rfloor )$ 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.