# On Gorenstein Circulant Graphs and Gorenstein SQC Graphs

## TL;DR

This paper characterizes Gorenstein properties of certain circulant and SQC graphs, providing specific conditions under which these graphs have Gorenstein edge ideals, thus advancing the understanding of algebraic properties linked to graph structures.

## Contribution

It offers a complete characterization of Gorenstein circulant graphs with bounded degree and of SQC graphs, identifying exact graph forms that satisfy the Gorenstein condition.

## Key findings

- Circulant graphs with degree ≤ 4 are Gorenstein iff they are tK_2, tC̅_n, or tC_{13}(1,5).
- SQC graphs are Gorenstein iff each component is an edge or a 5-cycle.
- Provides explicit classifications linking graph structure to Gorenstein property.

## 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.