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.

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.

Equations

$$\mathcal{A}=\{$$

$$(13d,2d,3d),(13d,4d,6d)|d\in \mathbb{N}\}.$$

$$(n,a,b)\in \mathcal{A}\cup \{(5d,d,2d),(8d,d,2d),(8d,2d,3d)|d\in \mathbb{N}\}.$$

$$\mathrm{V}(G)=\bigcup _{i=1}^m{S}_i\cup \bigcup _{i=1}^t\mathrm{V}(C_j)\cup \bigcup _{i=1}^r\mathrm{B}(Q_i)\text{hfill}$$

Full text

Keywords: Gorenstein simplicial complex; Edge ideal; Circulant graph; $\mathcal{SQC}$ graph.

2010 Mathematical Subject Classification: 13F55, 13H10, 05E40.

1 Introduction

Throughout this paper, $K$ is a field, $S=K[x_{1},\ldots,x_{n}]$ and $G$ denotes a simple undirected graph with vertex set $\mathrm{V}(G)=\{v_{1},\ldots,v_{n}\}$ and edge set $\mathrm{E}(G)$. Recall that the edge ideal $I(G)$ of $G$ is the ideal of $S$ generated by $\{x_{i}x_{j}|v_{i}v_{j}\in\mathrm{E}(G)\}$. Many researchers have studied how algebraic properties of $S/I(G)$ relates to the combinatorial properties of $G$ (see [14, 6, 8] and references therein). One of the algebraic properties that recently has gained attention is the property of being Gorenstein. We say that $G$ is a Gorenstein (resp, Cohen-Macaulay or CM for short) graph over $K$, if $S/I(G)$ is a Gorenstein (resp. CM) ring. When $G$ is Gorenstein (resp. CM) over every field, we say that $G$ is Gorenstein (resp. CM). In [7] and [8] a characterization of planar Gorenstein graphs of girth $\geq 4$ and triangle-free Gorenstein graphs, respectively, is presented. Also in [16] a condition on a planar graph equivalent to being Gorenstein is stated.

In this paper, first we recall some needed concepts and preliminary results. Then in Section 3, 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.

2 Preliminaries

Recall that a simplicial complex $\Delta$ on the vertex set $V=\{v_{1},\ldots,v_{n}\}$ is a family of subsets of $V$ (called faces) with the property that $\{v_{i}\}\in\Delta$ for each $i\in[n]=\{1,\ldots,n\}$ and if $A\subseteq B\in\Delta$, then $A\in\Delta$. In the sequel, $\Delta$ always denotes a simplicial complex. Thus the family $\Delta(G)$ of all cliques of a graph $G$ is a simplicial complex called the clique complex of $G$. Also $\Delta(\overline{G})$ is called the independence complex of $G$, where $\overline{G}$ denotes the complement of $G$. Note that the elements of $\Delta(\overline{G})$ are independent sets of $G$. The ideal of $S$ generated by $\{\prod_{v_{i}\in F}x_{i}|F\subseteq V$ is a non-face of $\Delta\}$ is called the Stanley-Reisner ideal of $\Delta$ and is denoted by $I_{\Delta}$ and $S/I_{\Delta}$ is called the Stanley-Reisner algebra of $\Delta$ over $K$. Therefore we have $I_{\Delta(\overline{G})}=I(G)$. Many researchers have studied the relation between combinatorial properties of $\Delta$ and algebraic properties of $S/I_{\Delta}$, see for example [6, 14, 1, 10, 9] and their references.

By the dimension of a face $F$ of $\Delta$, we mean $|F|-1$ and the dimension of $\Delta$ is defined as $\max\{\dim(F)|F\in\Delta\}$. Denote by $\alpha(G)$ the independence number of $G$, that is, the maximum size of an independent set of $G$. Note that $\alpha(G)=\dim\Delta(\overline{G})+1$. A graph $G$ is called well-covered, if all maximal independent sets of $G$ have size $\alpha(G)$ and it is said to be a W2 graph, if $|\mathrm{V}(G)|\geq 2$ and every pair of disjoint independent sets of $G$ are contained in two disjoint maximum independent sets. In some texts, W2 graphs are called 1-well-covered graphs. In the following lemma, we have collected some known results on W2 graphs and their relation to Gorenstein graphs.

Lemma 2.1**.**

1. (i)

(**[7, Lemma 3.1]** or **[8, Lemma 3.5]**)If $G$ is a graph without isolated vertices and $G$ is Gorenstein over some field $K$, then $G$ is a W2* graph.* 2. (ii)

(**[8, Proposition 3.7]**) If $G$ is triangle-free (that is, no subgraph of $G$ is a triangle) and without isolated vertices, then $G$ is Gorenstein if and only if $G$ is W2*.* 3. (iii)

(**[15, Theorem 4]**) The degree of every vertex of a connected non-complete W2* graph is at least 2.*

Recall that if $F\in\Delta$, then $\mathrm{link}_{\Delta}(F)=\{A\setminus F|F\subseteq A\in\Delta\}$. Suppose that $F\subseteq\mathrm{V}(G)$. By $\mathrm{N}[F]$ we mean $F\cup\{v\in\mathrm{V}(G)|uv\in\mathrm{E}(G)$ for some $u\in F\}$ and we set $G_{F}=G\setminus\mathrm{N}[F]$. Thus if $F$ is independent, then $\mathrm{link}_{\Delta(\overline{G})}F=\Delta(\overline{G_{F}})$. Also the polynomial $I(G,x)=\sum_{i=0}^{\alpha(G)}a_{i}x^{i}$, where $a_{i}$ is the number of independent sets of size $i$ in $G$, is called the independence polynomial of $G$. It should be mentioned that, if we set $f_{i}=a_{i+1}$ for $-1\leq i<\alpha(G)$, then the vector $(f_{-1},f_{0},\ldots,f_{\alpha(G)-1})$ is called the $f$-vector of $\Delta(\overline{G})$ in the literature of combinatorial commutative algebra.

Theorem 5.1 of [14] states conditions on $\Delta$ equivalent to the statement “$\Delta$ is Goresnstein over $K$”. Applying this theorem to the independence complex of $G$, we deduce the following characterization of Gorenstein graphs. The details of the proof can be found in [11].

Theorem 2.2**.**

Suppose that $G$ is a graph without isolated vertices. Then $G$ is Gorenstein over $K$ if and only if $G$ is CM over $K$, $I(G,-1)=(-1)^{\alpha(G)}$ and $\overline{G_{F}}$ is a cycle of length at least $4$ for each independent set $F$ of $G$ with size $\alpha(G)-2$. In particular, if $\alpha(G)=2$, then $G$ is Gorenstein if and only if $G$ is the complement of a cycle.

The above results can be applied to see that there are few Gorenstein graphs in many well-known classes of graphs. For example, let $G=K_{n}$ (the complete graph on $n$ vertices) for $n\geq 2$. Then $I(G,x)=1+nx$ and $\alpha(G)=-1$. Thus by (2.2), $G$ is Gorenstein if and only if $n=2$. Therefore, the only Gorenstein complete graphs are $K_{2}$ and $K_{1}$. Also it follows from [6, Corollary 9.1.14] and [4] that every CM bipartite graph and every CM very well-covered graph (that is, a well-covered graph $G$ with $|\mathrm{V}(G)|=2\alpha(G)$) has a vertex of degree 1, hence by (2.1), we see that the only connected Gorenstein bipartite or very well-covered graphs are $K_{2}$ and $K_{1}$. Also if $G$ is an $n$-cycle for $n\geq 3$, then $G$ is W2 only if $n=3$ or $n=5$ and it follows from (2.2), that the only Gorenstein cycle is $C_{5}$. In the next section, we search for Gorenstein graphs in two other classes of graphs, namely, $\mathcal{SQC}$ graphs and circulant graphs.

3 Gorenstein circulant and $\mathcal{SQC}$ graphs

In this section, we recall the definition of circulant and $\mathcal{SQC}$ graphs and characterize Gorenstein $\mathcal{SQC}$ graphs and Gorenstein circulant graphs with certain conditions. First, we consider circulant graphs. Suppose that $n\geq 3$ and $S\subseteq\{1,\ldots,\left\lfloor{n/2}\right\rfloor\}$. Then the circulant graph $C_{n}(S)$ is the graph with vertex set $[n]$ such that ${i,j}$ is an edge of $C_{n}(S)$ if and only if $\min\{|i-j|,n-|i-j|\}\in S$. In other words, assuming that $i>j$, the vertices $i$ and $j$ are adjacent if and only if $i-j\in S\cup(n-S)$, where by $n-S$ we mean $\{n-s|s\in S\}$. In [2], well-covered circulant graphs are studied and in [17], CM circulant graphs with degree 3 and CM circulant graphs of the form $C_{n}(1,\ldots,d)$ are characterized. Here we characterize Gorenstein circulant graphs with degree $\leq 4$ and Gorenstein circulant graphs of the form $C_{n}(1,\ldots,d)$.

Theorem 3.1**.**

Assume that $n\geq 3$, $d\leq\left\lfloor{n/2}\right\rfloor$ and $K$ is an arbitrary field and let $G=C_{n}(1,\ldots,d)$. Then $G$ is Gorenstein (over $K$) if and only if $n=2d+3$.

Proof.

If $\alpha(G)=1$, then $G$ is a complete graph. But no complete graph on more than two vertices is Gorenstein. So $\alpha(G)\geq 2$. Note that each vertex of $G$ is adjacent exactly to the (cyclically) next and previous $d$ vertices. Thus the set $A_{t}=\{1,d+2,2d+3,\ldots,1+t(d+1)\}$ is an independent set of $G$, if $1+t(d+1)\leq n-d$ and is a maximal independent set if $t=t_{0}=\left\lfloor{(n-d-1)/(d+1)}\right\rfloor$. Since $G$ is well-covered, $t_{0}=\alpha(G)$. Assume that $\alpha(G)\geq 3$. Set $x=1+t_{0}(d+1)$ and $F=A_{t_{0}}\setminus\{1,x\}$. Then by (2.2) $\overline{G_{F}}$ is a cycle, that is each vertex of $G_{F}$ is adjacent to all except two other vertices of $G_{F}$. Note that $G_{F}$ is the induced subgraph of $G$ on vertices $x,x+1,\ldots,n,1$. Since the vertices of $G_{F}$ which are not adjacent to 1 are $x,x+1,\ldots,n-d$, we conclude that $n-d=x+1$ (see (Fig. 1)). But then $x+1$ is adjacent to all vertices of $G_{F}$, except 1 which is a contradiction. Consequently, $\alpha(G)=2$ and by (2.2), $\bar{G}$ is a cycle. Thus degree of each vertex of $G$ which is $2d$ must equal $n-3$, that is, $n=2d+3$ as required. Conversely, if $n=2d+3$, then $G$ is the complement of cycle and hence Gorenstein. ∎

Now we are going to characterize Gorenstein circulant graphs with vertex degree $\leq 4$. Note that the only circulant graphs with degree $1$ are $C_{2n}(n)$ which are disjoint unions of a set of edges and hence are Gorenstein. Also connected circulant graphs with vertex degree $2$ are exactly the cycles $C_{n}$ which are Gorenstein if and only if $n=5$. In [17], CM graphs with vertex degree 3 are characterized. Using their results we get the following. Note that Gorenstein graphs with vertex degree $3$ are $C_{2n}(a,n)$ for some $1\leq a<n$.

Theorem 3.2**.**

The circulant graph $G=C_{2n}(a,n)$ with $1\leq a<n$ is Gorenstein if and only if $2n/\gcd(a,2n)=3$. In this case, $G$ is isomorphic to a disjoint copies of complement of a 6-cycle.

Proof.

Let $t=\gcd(a,2n)$ and assume that $G$ is Gorenstein. Since $G$ is CM and according to [17, Theorem 5.5], we get that $2n/t=3$ or $2n/t=4$. If $2n/t=4$, then by [17, Theorem 5.3], $G$ is isomorphic to $t$ copies of $C_{4}(1,2)=K_{4}$ which is not Gorenstein. So $2n/t=3$. Conversely, if $2n/t=3$, then by [17, Theorem 5.3], $G$ is isomorphic to $t/2$ copies of $C_{6}(2,3)=\overline{C_{6}}$ which is Gorenstein according to (2.2). ∎

It remains to characterize circulant graphs with vertex degree equal to 4. These graphs are $C_{n}(a,b)$ with $1\leq a<b<n/2$. For this, we need several lemmas. In the sequel, we always assume that $1\leq a<b<n/2$. It should be mentioned that in what follows we have checked that some specific graphs are Gorenstein or W2 or …, using the computer algebra system Macaulay 2 [5].

Lemma 3.3**.**

Suppose that $B$ is an independent set of $G$ and $v\in B$. If $B\cap\mathrm{N}_{G}(x)\neq\{v\}$ for each $x\in\mathrm{N}_{G}(v)$, then $G$ is not W2.

Proof.

Suppose that $G$ is W2. By replacing $B$ with a maximum independent set of $G$ containing $B$, we can assume that $B$ is an independent set of size $\alpha(G)$. Let $A_{1}=\{v\}$ and $A_{2}=B\setminus\{v\}$. Since $G$ is W2, there are disjoint maximum independent sets $B_{1}$ and $B_{2}$ containing $A_{1}$ and $A_{2}$, respectively. Assume that $B_{2}=A_{2}\cup\{x\}$. As $x\notin B_{1}$, we have $x\neq v$. If $x$ is not adjacent to $v$, then since $B$ is a maximum independent set of $G$, $x$ is adjacent to some vertex of $B\setminus\{v\}=A_{2}\subseteq B_{2}$, which contradicts independency of $B_{2}$. Thus $x$ is adjacent to $v$. Hence by assumption, there is a vertex $y\in B\cap\mathrm{N}_{G}(x)$, with $y\neq v$. Again $y\in A_{2}\subseteq B_{2}$ and we get a contradiction. Consequently, $G$ is not W2. ∎

Lemma 3.4**.**

Suppose that $G=C_{n}(a,b)$ and $d|n$ $d|a$ and $d|b$. Then $G$ is isomorphic to $d$ disjoint copies of $C_{n/d}(a/d,b/d)$.

Proof.

For each $0\leq i<d$, let $V_{i}=\{v\in\mathrm{V}(G)|v\equiv i\mod{d}\}$. Note that for each such $i$, we have $|(kd+i)-(k^{\prime}d+i)|\in\{a,b\}$ if and only if $|(k-k^{\prime})|\in\{a/d,b/d\}$. Thus the induced subgraph of $G$ on each $V_{i}$ is isomorphic to $C_{n/d}(a/d,b/d)$ and it is clear that there is no edge in $G$ between $V_{i}$ and $V_{j}$ for $i\neq j$ and the result follows. ∎

Note that when $\gcd(n,a)=1$, then $a$ has a multiplicative inverse modulo $n$, which we denote by $a^{-1}$.

Lemma 3.5**.**

If $\gcd(n,a)=1$ (resp. $\gcd(n,b)=1$), then $C_{n}(a,b)$ is isomorphic to $C_{n}(1,d)$ with $d=\min\{a^{-1}b\mod{n},-a^{-1}b\mod{n}\}$ (resp. $d=\min\{b^{-1}a\mod{n},-b^{-1}a\mod{n}\}$).

Proof.

Assume that $\gcd(n,a)=1$ (the proof for $\gcd(n,b)=1$ is similar). So $a$ has an inverse modulo $n$ and $d$ is well-defined and the “$\min$” in the definition of $d$ guarantees $d<n/2$ . For each $i\in\mathrm{V}(C_{n}(1,d))$ set $\phi(i)=1+(i-1)a\mod{n}$ and consider $\phi$ as a map $C_{n}(1,d)\to C_{n}(a,b)$. If for $1\leq i,i^{\prime}\leq n$ we have $\phi(i)=\phi(i^{\prime})$, then $i\equiv i^{\prime}\mod{n}$ (because $a$ is invertible modulo $n$) and hence $i=i^{\prime}$. Thus $\phi$ is one-to-one and onto. Also it is easy to verify that $\phi(i)$ and $\phi(i+1)$ and also $\phi(i)$ and $\phi(i+d)$are adjacent in $C_{n}(a,b)$ (where we compute $i+1$ or $i+d$ modulo $n$, if $i+1>n$ or $i+d>n$, respectively). Since the two graphs have the same number of edges, it follows that $\phi$ is an isomorphism. ∎

Lemma 3.6**.**

For $n\geq 4$, the graph $G=C_{n}(1,2)$ is W2 if and only if $n\in\{4,5,7,8\}$.

Proof.

Suppose that $G$ is W2 and hence well-covered. By [2, Theorem 4.1] we have $n\leq 8$ or $n=11$. One can check that for $n=6$ or $11$, $G$ is not W2 and for the other possible $n$’s, $G$ is W2. ∎

Now we can characterize Gorenstein and W2 circulant graphs of degree 4.

Theorem 3.7**.**

Suppose that $1\leq a<b<n/2$. The graph $C_{n}(a,b)$ is Gorenstein (over $K$) if and only if $(n,a,b)\in\mathcal{A}$ where

[TABLE]

Also $C_{n}(a,b)$ is W2 if and only if

[TABLE]

Proof.

Assume that $G$ is W2. By using (3.4), we assume that $\gcd(n,a,b)=1$ (which corresponds to $d=1$). First we prove the following claim.

Claim.

One of the following relations hold: $n=2a+b$ or $n=a+2b$ or $n=2a+3b$ or $n=3a+2b$ or $2n=2a+3b$ or $2n=3a+2b$.

Proof of Claim.

Let $B=\{1,1+a+b,n-a-b+1\}$. Note that vertices $1+a$ and $1+b$ are adjacent to $1+a+b$ and vertices $n-a+1$ and $n-b+1$ are adjacent to $n-a-b+1$, thus every neighbor of $1$ has a neighbor in $B\setminus\{1\}$. If $B$ is independent, then by (3.3) with $v=1$, we deduce that $G$ is not W2. Therefore $B$ is not independent. Suppose that $1+a+b\in\mathrm{N}(1)$. Then as $1+a+b>1$, we should have either $a+b\in\{a,b\}$ (which is not possible) or $a+b\in\{n-a,n-b\}$. The latter condition is equivalent to either $n=2a+b$ or $n=a+2b$. Similarly if $n-a-b+1\in\mathrm{N}(1)$, then again either $n=2a+b$ or $n=a+2b$.

Now assume that $1+a+b\in\mathrm{N}(n-a-b+1)$. If $1+a+b<n-a-b+1$, then $n-2a-2b\in\{a,b,n-a,n-b\}$ which implies that either $n=2a+3b$ or $n=3a+2b$. Finally if $n-a-b+1<1+a+b$, then we must have $2a+2b-n\in\{a,b,n-a,n-b\}$, that is, either $n=a+2b$ or $n=2a+b$ or $2n=3a+2b$ or $2n=2a+3b$. This concludes the proof of the claim. Now we consider several cases.

Case 1: $n=2a+b$* or $n=a+2b$.*

Assume that $n=a+2b$ and let $d=\gcd(n,b)$. Then $d|a$ and hence $d=\gcd(n,a,b)=1$. Note that $a\equiv 2b\mod{n}$ and hence $b^{-1}a\equiv 2\mod{n}$. Thus according to (3.5), $G$ is isomorphic to $C_{n}(1,2)$. Also if $n=2a+b$, a similar argument shows that $G\cong C_{n}(1,2)$. Note that since degree of each vertex in $G$ is 4, $n\geq 5$. Therefore, it follows from (3.6), that $n\in\{5,7,8\}$. Consequently, in this case $G$ is W2 if and only if $(n,a,b)$ equals one of the following: $(5,1,2)$, $(7,1,3)$, $(7,2,3)$ or $(8,2,3)$. Moreover, according to (3.1), in this case $G$ is Gorenstein if and only if $n=7$.

Case 2: $n=2a+3b$.

Consider $B=\{1,1+2a,1+b-a,n-a-b+1\}$. Every neighbor of 1 has a neighbor in $B\setminus\{1\}$. Hence by (3.3), $B$ is not independent. If $1+2a\in\mathrm{N}(1)$, then $2a\in\{a,b,n-a,n-b\}$. The only possible case is that $2a=b$ (for example, if $2a=n-a$, then by replacing $n$ with $2a+3b$, it follows that $a=3b$, which contradicts $a<b$). But then $n=8a$ and hence $a=\gcd(n,a,b)=1$, that is, $(n,a,b)=(8,1,2)$. Note that $C_{8}(1,2)$ is W2 but not Gorenstein, according to (3.1) and (3.6). If $1+b-a\in\mathrm{N}(1)$, then again the only possible case is $b-a=a$ which leads to $(n,a,b)=(8,1,2)$ as above. Also note that if $n-a-b+1\in\mathrm{N}(1)$, then by the proof of the claim above, $n=2a+b$ or $n=a+2b$ which contradicts the assumption of this case.

Now assume $1+2a$ and $1+b-a$ are adjacent. If $1+b-a>1+2a$, then we have $b-3a\in\{a,b,n-a,n-b\}$. But the only possible case is $b-3a=a$, or equivalently $b=4a$. Similar to the case $b=2a$, we see that $a=\gcd(n,a,b)$=1 and $(n,a,b)=(14,1,4)$. But $C_{14}(1,4)$ is not W2. If $1+b-a<1+2a$, then we have $3a-b\in\{a,b,n-a,n-b\}$. It follows that either $b=2a$ or $2b=3a$. The former case was dealt above. In the latter case, we have $b=3t$ for some integer $t$ and $a=2t$, hence $n=13t$ and $t=\gcd(n,a,b)=1$, that is, $(n,a,b)=(13,2,3)$. Using Macaulay 2, we see that $C_{13}(2,3)$ is W2. Also no two neighbors of 1 in $C_{13}(2,3)$ are adjacent, which means, 1 is not in any triangle of $C_{13}(2,3)$. But circulant graphs are vertex transitive and hence $C_{13}(2,3)$ is triangle-free and Gorenstein by (2.1)(ii).

Finally, noting that $\max\{1+2a,1+b-a\}<n-b-a+1=a+2b+1$, we see that $\{1+2a,1+b-a\}\cap\mathrm{N}(n-b-a+1)\neq\emptyset$ if and only if $2a+b$ or $2b-a$ are in $\{a,b,n-a,n-b\}$. But all of these cases lead to contradictions with $1\leq a<b<n/2$ and are not possible. Consequently, under the assumption of case 2, $G$ is W2 if and only if $(n,a,b)$ is $(8,1,2)$ or $(13,2,3)$ and only in the latter case $G$ is Gorenstein.

Case 3: $n=3a+2b$.

Consider $B=\{1,1+b-a,1+b+a,n-b+a+1\}$. By (3.3), $B$ is not independent. Note that $v_{1}=1<v_{2}=1+b-a<v_{3}=1+b+a<v_{4}=n-b+a+1=1+4a+b$. Thus there are some $i$ and $j$ such that $1\leq i<j\leq 4$ and $v_{j}-v_{i}\in\{a,b,n-a,n-b\}$. Writing down all of these equations for $1\leq i<j\leq 4$ and ruling out those which contradict $1\leq a<b<n/2$, we deduce that either $b=2a$ or $b=3a$ or $b=5a$ or $2b=3a$. Similar to the above argument, in the first three cases, it follows that $a=\gcd(n,a,b)=1$ and in the last case $a/2=b/3=\gcd(n,a,b)=1$. Therefore, $(n,a,b)$ is one of the following: $(7,1,2)$ or $(9,1,3)$ or $(13,1,5)$ or $(12,2,3)$. Using Macaulay 2, we see that $G$ is W2 only in the cases $(n,a,b)\in\{(7,1,2),(13,1,5)\}$. Also $C_{7}(1,2)$ is Gorenstein by (3.1) and $C_{13}(1,5)$ is Gorenstein, because it is triangle-free and W2.

Case 4: $2n=2a+3b$.

First note that in this case $b$ is even and we have $b<2a$ (else $n\leq 3b/2+b/2=2b$, contradicting $b<n/2$). Let $B=\{1,1+b-a,n-b-a+1=1+b/2,1+2a\}$. By (3.3), $B$ is not independent and hence the difference of two of the elements of $B$ should be in $\{a,b,n-b,n-a\}$. The equations that can be derived from this point and not contradicting $1\leq a<b<\min\{n/2,2a\}$ are: $3a=2b$, $4a=3b$ or $6a=5b$. Assume that $3a=2b$, then $a=2t$ and $b=3t$. Since $b$ is even, $t=2t^{\prime}$ and hence $n=a+3b/2=13t^{\prime}$. Since $t^{\prime}=\gcd(n,a,b)=1$, we deduce that $(n,a,b)=(13,4,6)$. By a similar argument in other cases we have $(n,a,b)=(9,3,4)$ or $(14,5,6)$. Neither $C_{14}(5,6)$ nor $C_{9}(3,4)$ is W2. Consequently, $G$ is W2 if and only if $(n,a,b)=(13,4,6)$ and if this is the case, then $G$ is also Gorenstein, because $C_{13}(4,6)$ is triangle-free.

Case 5: $2n=3a+2b$.

In this case, $a$ is even and $b<3a/2$. Let $B=\{1<1+b-a<1+n-b-a=1+a/2<1+5a/2\}$. Note that $(1+5a/2)-(1+a)=n-b$ and hence $1+a$ is adjacent to $1+5a/2$. Similarly, each other neighbor of $1$ is adjacent to an element of $B\setminus\{1\}$. If $B$ is not independent, then the difference of two of its members should be in $\{a,b,n-a,n-b\}$. But all of the equations that follow from this, contradict $1\leq a<b<\min\{3a/2,n/2\}$. Hence $B$ is independent and by (3.3), $G$ is not W2, that is, this case is not possible. This completes the proof of the theorem. ∎

We can simplify the above theorem, noting that several of the graphs in that theorem are isomorphic.

Corollary 3.8**.**

Let $G$ be a circulant graph with vertex degree $4$. Then $G$ is W2 if and only if $G$ is isomorphic to disjoint copies of one of the following graphs: $C_{7}(1,2),C_{13}(1,5),C_{5}(1,2),C_{8}(1,2)$. Moreover, $G$ is Gorenstein if and only if it is isomorphic to a disjoint copies of $C_{7}(1,2)$ or to a disjoint copies of $C_{13}(1,5)$.

Proof.

Note that using (3.5), we get $C_{13}(2,3)\cong C_{13}(4,6)\cong C_{13}(1,5)$, $C_{7}(1,2)\cong C_{7}(2,3)\cong C_{7}(1,3)$ and $C_{8}(2,3)\cong C_{8}(1,2)$. Now the result follows from (3.7) and (3.4). ∎

Suppose that $H$ is a graph and $t\in\mathbb{N}$. By $tH$ we mean the graph consisting of $t$ disjoint copies of $H$. Summarizing our results on Gorenstein circulant graphs, we get the following.

Corollary 3.9**.**

Suppose that $G$ is a circulant graph with vertex degree $\leq 4$ 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 $t\in\mathbb{N}$ and $4\leq n\in\mathbb{N}$.

Next we consider $\mathcal{SQC}$ graphs. Our interest in $\mathcal{SQC}$ graphs arise form the fact that these graphs are CM and also every CM graph with girth at least 5 or every CM graphs which does not contain any cycles of length 4 or 5 is in the class $\mathcal{SQC}$, see [7]. We recall the definition of $\mathcal{SQC}$ graphs from [13]. By a simplicial vertex of $G$, we mean a vertex $v$ such that $\mathrm{N}_{G}[v]$ is complete and in this case we call $\mathrm{N}_{G}[v]$ a simplex of $G$. Furthermore, a 5-cycle $C$ of $G$ is called a* basic 5-cycle* when $C$ does not contain two adjacent vertices both with degree more than 2. Also a 4-cycle $C$ is said to be* basic*, if it contains two adjacent vertices of degree 2 and the two other vertices of $C$ each belong to a simplex or to a basic 5-cycle. For a basic 4-cycle $C$ we denote the set of degree 2 vertices of $C$ by $\mathrm{B}(C)$ and call them basic vertices of $C$. If there are simplices $S_{1},\ldots,S_{m}$, basic 5-cycles $C_{1},\ldots,C_{t}$ and basic 4-cycles $Q_{1},\ldots,Q_{r}$ of $G$ such that

[TABLE]

is a partition of $\mathrm{V}(G)$, we say that $G$ is in the class $\mathcal{SQC}$ (or simply, $G$ is $\mathcal{SQC}$) and call ($*$ ‣ 3) a $\mathcal{SQC}$ partition of $\mathrm{V}(G)$. Note that in this case, $G$ is well-covered and has independence number equal to $m+2t+r$ (see [13, Theorem 3.1]).

Theorem 3.10**.**

Suppose that $G$ has no isolated vertex and is in the class $\mathcal{SQC}$. Then $G$ is Gorenstein if and only if each component of $G$ is either an edge or a 5-cycle.

Proof.

Since $G$ is Gorenstein if and only if all of its components are so, we assume that $G$ is connected. Suppose that ($*$ ‣ 3) is a $\mathcal{SQC}$ partition of $\mathrm{V}(G)$. First note that $r=0$, since by [12, Theorem 2] every vertex of a 4-cycle of $G$ has degree at least 4 and $G$ cannot have any basic 4-cycles. Suppose that $S_{i}=\mathrm{N}[x_{i}]$. If $\deg(x_{i})=1$ for some $i$, then it follows from (2.1) that $G$ is just an edge. Thus assume that $\deg(x_{i})\geq 2$ for each $i$.

Suppose that $m>1$. Note that in each basic 5-cycle of an arbitrary graph, there exist at least two non-adjacent vertices of degree two. Let $F$ be the independent set consisting of $x_{3},\ldots,x_{m}$ and two non-adjacent vertices of degree two from each $C_{i}$. Then $|F|=\alpha(G)-2$ and by (2.2) $\overline{G_{F}}$ is a cycle. But as $x_{1}$ is not adjacent to $N_{G}[x_{2}]$ in $G$ and $N_{G}[x_{2}]\subseteq\mathrm{V}(G_{F})$, we see that $\deg_{\overline{G_{F}}}(x_{1})\geq|\mathrm{N}_{G}[x_{2}]|\geq 3$, which is a contradiction. If $m=1$, then $t>0$, else $G$ is a complete graph on at least 3 vertices and is not Gorenstein. Let $F$ be the independent set consisting of one degree 2 vertex of $C_{1}$, say $y$, and two non-adjacent vertices of degree 2 from each $C_{j}$ with $j>1$. Again $\overline{G_{F}}$ should be a cycle but $deg_{\overline{G_{F}}}(y)\geq|\mathrm{N}_{G}[x_{1}]|\geq 3$. Therefore, we conclude that $m=0$.

Now suppose that $t\geq 2$. Because $G$ is connected, there is an edge between two vertices of two of the $C_{i}$’s, say vertices $y_{1}$ and $y_{2}$ of $C_{1}$ and $C_{2}$, respectively. At least one of the vertices of $C_{1}$ which are not adjacent to $y_{1}$ has degree 2, because $C_{1}$ is basic; call this vertex $y^{\prime}_{1}$. Let $F$ to be the independent set containing $y^{\prime}_{1},y_{2}$ and two non-adjacent vertices of degree 2 from each of the $C_{i}$ for $i>2$. Then $G_{F}$ is the disjoint union of an edge and an isolated vertex which is not the complement of a cycle. This contradicts (2.2) and we conclude that $t=1$. Consequently, either $G=C_{1}$ is a 5-cycle, or $G$ is obtained from $C_{1}$ by adding one edge. In the latter case, by applying (2.2), we see that $G$ is not Gorenstein and the result follows. ∎