On Gorenstein Circulant Graphs and Gorenstein SQC Graphs

Abstract

We characterize some graphs with a Gorenstein edge ideal. In particular, we show that if $G$ is a circulant graph with vertex degree at most four or a circulant graph of the form $C_n(1,\ldots, d)$ for some $d\leq n/2$, then $G$ is Gorenstein if and only if $G\cong tK_2$, $G\cong t\overline{C_n}$ or $G\cong tC_{13}(1,5)$ for some integers $t$ and $n\geq 4$. Also we prove that if $G$ is a \mathcal{SQC}\ graph, then $G$ is Gorenstein if and only if each component of $G$ is either an edge or a 5-cycle.

Figures (1)

• Figure 1: The graphs $G$ and $G_F$ in the proof of (\ref{['Cn(1..d)']}); the dashed part is not in $G_F$

Theorems & Definitions (21)

• Lemma 2.1
• Theorem 2.2
• Theorem 3.1
• proof
• Theorem 3.2
• proof
• Lemma 3.3
• proof
• Lemma 3.4
• proof