On the reduced Euler characteristic of independence complexes of circulant graphs

TL;DR

This paper investigates the reduced Euler characteristic of independence complexes of specific circulant graphs, proving non-vanishing in certain cases and providing a counterexample to a conjecture.

Contribution

It establishes non-zero reduced Euler characteristic for circulant graphs with n=p^k and n=2p^k, and presents a counterexample to a previous conjecture.

Findings

Proved non-vanishing of e for n=p^k and n=2p^k.

Provided a circulant graph with e=0, countering prior assumptions.

Enhanced understanding of topological invariants of independence complexes.

Abstract

Let $G$ be the circulant graph $C_n(S)$ with $S\subseteq\{ 1,\ldots,\left \lfloor\frac{n}{2}\right \rfloor\}$. We study the reduced Euler characteristic $\tilde{\chi}$ of the independence complex $\Delta (G)$ for $n=p^k$ with $p$ prime and for $n=2p^k$ with $p$ odd prime, proving that in both cases $\tilde{\chi}$ does not vanish. We also give an example of circulant graph whose independence complex has $\tilde{\chi}$ equals to $0$, giving a negative answer to R. Hoshino.

Tables

Table 1. Table 1. The table shows G = C n ​ ( S ) 𝐺 subscript 𝐶 𝑛 𝑆 G=C_{n}(S) such that χ ~ ​ ( Δ ​ ( G ) ) = 0 ~ 𝜒 Δ 𝐺 0 \widetilde{\chi}(\Delta(G))=0 , up to isomorphisms.

	$n=30$
$\{1,3,8\}$	$\{1,7,9,11,14\}$	$\{1,2,3,7,9,11,13\}$
$\{2,9,13\}$	$\{1,4,9,13,14\}$	$\{2,3,4,5,7,9,14\}$
$\{8,9,13\}$	$\{2,3,7,8,9\}$	$\{2,3,4,5,8,9,14\}$
$\{1,8,9,14\}$	$\{1,3,4,9,11\}$	$\{1,3,4,5,7,8,14\}$
$\{2,3,11,13\}$	$\{2,7,8,9,13\}$	$\{2,3,4,5,8,11,13\}$
$\{3,8,11,13\}$	$\{2,3,4,7,13\}$	$\{1,2,3,7,8,9,11,13\}$
$\{1,3,4,13\}$	$\{1,3,4,5,7,8\}$	$\{2,3,5,8,9,11,13,14\}$
$\{7,8,9,13\}$	$\{2,3,4,5,8,11\}$	$\{1,2,3,4,5,8,9,14\}$
$\{1,4,7,9\}$	$\{1,2,3,8,9,11\}$	$\{1,2,3,5,7,9,11,14\}$
$\{1,8,9,11\}$	$\{1,3,4,7,9,13\}$	$\{2,3,4,5,7,9,13,14\}$
$\{2,9,11,14\}$	$\{1,4,7,9,11,14\}$	$\{1,3,4,5,7,8,9,11,13\}$
$\{1,2,9,13\}$	$\{1,2,3,5,11,14\}$	$\{1,2,3,4,5,8,9,11,13\}$
$\{2,3,7,9\}$	$\{1,3,4,9,11,14\}$	$\{2,3,4,5,7,8,9,13,14\}$
$\{1,7,8,9,11\}$	$\{2,3,4,7,8,13\}$	$\{1,2,3,4,5,7,9,11,13,14\}$
$\{1,3,7,8,13\}$	$\{1,2,5,7,9,13,14\}$
$\{2,3,4,7,8\}$	$\{1,4,5,7,8,9,11\}$
	$n=36$
$\{2,3,6,7,10,14,15\}$	$\{2,5,6,7,10,11,14\}$	$\{2,5,6,10,11,13,14\}$
$\{1,2,5,6,7,10,11,17\}$	$\{1,5,6,7,11,13,14,17\}$	$\{2,5,6,7,10,14,15,17\}$
$\{1,2,5,6,7,10,11,13\}$	$\{1,5,6,7,10,11,13,14,17\}$

Equations

$$I(G,x)=\sum _{i=0}^n{f}_{i-1}{x}^i,$$

$$I(G)=(x_i{x}_j:\{i,j\}\in E(G)).$$

$$h_k=\sum _{i=0}^k(-1{)}^{k-i}(\genfrac{}{}{0ex}{}{d-i}{k-i})f_{i-1}.$$

$$\tilde{\chi }(\Delta )=\sum _{i=0}^d(-1{)}^{i-1}{f}_{i-1}$$

$$I_{\Delta }=(\{x_{j_1}{x}_{j_2}\cdots x_{j_r}:\{x_{j_1},x_{j_2},\dots ,x_{j_r}\}\notin \Delta \}).$$

$$\Delta (G)=\{A\subset V(G):A\text{mbox}isanindependentsetofG\}.$$

$$\tilde{\chi }(\Delta (G))=-I(G,-1)$$

$$H_{R/I}(k):={\dim }_K(R/I{)}_k$$

$${\mathrm{HP}}_{R/I}(t):=\sum _{k\in \mathbb{N}}{H}_{R/I}(k)t^k.$$

$${\mathrm{HP}}_{R/I}(t)=\frac{h(t)}{(1-t{)}^n}.$$

$$\mathrm{ri}(R/I_{\Delta})=\left\{\begin{tabular}[]{cl}$0$&if $h_{d}=0$\\ $1$&if $h_{d}\neq 0$.\end{tabular}\right.$$

$$\begin{gathered} i\cdot f_{i-1}=n\cdot f_{i-1,0} \\ \text{mbox}with \\ 0\le i\le d. \end{gathered}$$

$${\mathcal{F}}_{i-1}=\{F_1,\dots ,F_{f_{i-1}}\}.$$

$$a_{jk}=\{\begin{array}{ll}1& \text{mbox}ifthevertexk-1\in F_j\\ 0& \text{mbox}otherwise.\end{array}$$

$$\omega(G)\leq\Big{\lfloor}\dfrac{n}{2}\Big{\rfloor}.$$

$$S=\Big{\{}1,2,\ldots,\Big{\lfloor}\frac{n}{2}\Big{\rfloor}\Big{\}},$$

$$\begin{gathered} i\cdot f_{i-1}=p^k\cdot f_{i-1,0} \\ \text{mbox}with \\ 0\le i\le d. \end{gathered}$$

$$\tilde{\chi }(\Delta (G))=\sum _{i=1}^d(-1{)}^{i-1}{f}_{i-1}-1=pr-1$$

$$V_1=\{2k|k=0,\dots ,q-1\}\text{mbox}and{V}_2=\{2k+1|k=0,\dots ,q-1\}.$$

$$V^{\prime }=\{v_1,v_2,\dots ,v_{\frac{q-1}{2}+1}\},$$

$$V_i=\{rk+i|k=0,\dots ,q-1\}$$

$$f_{p^k-1}\equiv 2modp.$$

$$f_{p^k-1}=\left(\genfrac{}{}{0ex}{}{2p^k}{p^k}\right)\equiv 2modp.$$

$$V_1=\{0,2,4,6,\dots ,2p^k-2\}\text{mbox}and{V}_2=\{1,3,5,7,\dots ,2p^k-1\}.$$

$$2bt\in V\text{mbox}and(2b+1)t\in V^{\prime }\text{mbox}foranyb.$$

$$\Big{\{}t:t\mbox{ odd and }(t,p)=1\Big{\}}\cup\Big{\{}2k:k=1,\ldots,\frac{p^{k}-1}{2}\Big{\}}\subseteq S.$$

$$(1+2x{)}^{p^k}$$

$$V=V+j,$$

$$L=\{0,b,2b,\dots (o(b)-1)b\}\subseteq V.$$

$$V,V+1,\dots ,V+(2sp-1)$$

Full text

Keywords. Circulant graph, Euler Characteristic, Simplicial Complex.

Introduction

Let $G$ be a finite simple graph with vertex set $V(G)$ and edge set $E(G)$. A subset $C$ of $V(G)$ is a clique of $G$ if any two different vertices of $C$ are adjacent in $G$. A subset $A$ of $V(G)$ is called an independent set of $G$ if no two vertices of $A$ are adjacent in $G$. The complement graph of $G$, $\bar{G}$, is the graph with vertex set $V(G)$ and edge set $E(\bar{G})=\{\{u,v\}\mbox{ with }u,v\in V(G)\mid\{u,v\}\notin E(G)\}$. In particular, a set is independent in $G$ if and only if it is a clique in the complement graph $\bar{G}$.

We also recall that a circulant graph is defined as follows. Let $S\subseteq\{1,2,\ldots,\left\lfloor\frac{n}{2}\right\rfloor\}$. The circulant graph $G:=C_{n}(S)$ is a simple graph with $V(G)=\mathbb{Z}_{n}=\{0,\ldots,n-1\}$ and $E(G):=\{\{i,j\}\mid|j-i|_{n}\in S\}$ where $|k|_{n}=\min\{|k|,n-|k|\}$.

Recently many authors have studied some combinatorial and algebraic properties of circulant graphs (see [7], [3], [2], [12], [5], [10]). In particular, in [7], [3], [2], [5], a formula for the $f$-vector of the independence complex was showed for some nice classes of circulants, e.g. the $d$-th power cycle, $S=\{1,2,\ldots,d\}$, and its complement. Moreover, Hoshino in [7, p. 247] focused on the Euler characteristic, an invariant that is associated to any simplicial complex (see [4]). In particular, he conjectured, by our notation, that any independence complex associated to a non-empty circulant graph has reduced Euler characteristic always different from [math].

We show that for particular $n$, a circulant graph $C_{n}(S)$ will support the conjecture, independent of the choice on $S$. To this aim, we exploit that each entry of the $f$-vector is a multiple of a divisor of $n$ (see Remark 2.1).

In Section 2 we prove that the conjecture holds for $n=p^{k}$ for any prime $p$, and for $n=2p^{k}$ for any odd prime $p$. Moreover we disprove the conjecture providing a counterexample (see Example 2.10).

As an application of our results, we focus our attention on two algebraic objects related to the independence complex of circulant graphs. We first consider the independence polynomial (see [7],[2]), that is

[TABLE]

where $f_{i-1}$ are the entries of the $f$-vector of the independence complex of a graph $G$. In particular, $-1$ is a root of the independence polynomial if and only if the Euler characteristic of the independence complex vanishes. This happens in Example 2.10 and does not happen for all the cases studied in Theorems 2.3, 2.9.

The second one arises from commutative algebra (see also [4], [9], [14], [11]). Let $R=K[x_{0},\ldots,x_{n-1}]$ be the polynomial ring and $I(G)$ the edge ideal related to the graph $G$ (see [13]), that is

[TABLE]

In this case the non-vanishing of the reduced Euler charateristic gives us information about the regularity index of $R/I(G)$, namely the smallest integer such that the Hilbert function on $R/I(G)$ becomes a polynomial function, the so-called Hilbert polynomial (see Section 1, Remark 1.2). Also in this case Theorems 2.3, 2.9 and Example 2.10 are relevant.

1. Preliminaries

In this section we recall some concepts and notations on graphs and on simplicial complexes that we will use in the article.

Set $V=\{x_{1},\ldots,x_{n}\}$. A simplicial complex $\Delta$ on the vertex set $V$ is a collection of subsets of $V$ such that: 1) $\{x_{i}\}\in\Delta$ for all $x_{i}\in V$; 2) $F\in\Delta$ and $G\subseteq F$ imply $G\in\Delta$. An element $F\in\Delta$ is called a face of $\Delta$. A maximal face of $\Delta$ with respect to inclusion is called a facet of $\Delta$.

The dimension of a face $F\in\Delta$ is $\dim F=|F|-1$, and the dimension of $\Delta$ is the maximum of the dimensions of all facets. Let $d-1$ be the dimension of $\Delta$ and let $f_{i}$ be the number of faces of $\Delta$ of dimension $i$ with the convention that $f_{-1}=1$. Then the $f$-vector of $\Delta$ is the $(d+1)$-tuple $f(\Delta)=(f_{-1},f_{0},\ldots,f_{d-1})$. The $h$-vector of $\Delta$ is $h(\Delta)=(h_{0},h_{1},\ldots,h_{d})$ with

[TABLE]

The sum

[TABLE]

is called the reduced Euler characteristic of $\Delta$ and $h_{d}=(-1)^{d-1}\widetilde{\chi}(\Delta)$.

Given any simplicial complex $\Delta$ on $V$, we can associate a monomial ideal $I_{\Delta}$ in the polynomial ring $R$ as follows:

[TABLE]

$R/I_{\Delta}$ is called Stanley-Reisner ring and its Krull dimension is $d$. If $G$ is a graph, the independence complex of $G$ is

[TABLE]

The independence polynomial is associated to $\Delta(G)$ and by Equation (0.1) it follows

[TABLE]

We also remark that from the definition of Stanley-Reisner ring and by Equation (0.2), it follows $R/I_{\Delta(G)}=R/I(G)$.

The clique complex of a graph $G$ is the simplicial complex whose faces are the cliques of $G$.

Remark 1.1**.**

Let $G=C_{n}(S)$ be a circulant graph on $S\subseteq T:=\{1,2,\ldots,\left\lfloor\frac{n}{2}\right\rfloor\}$. We observe that the complement graph of $G$, namely $\bar{G}$, is a circulant graph on $\bar{S}:=T\setminus S$. Moreover the clique complex of $\bar{G}$ is the independence complex of $G$, $\Delta(G)$.

We also recall some basic facts about the regularity index (see [14, Chapter 5]). Let $R$ be a standard graded ring and $I$ be a homogeneous ideal. The Hilbert function $\operatorname{H}_{R/I}:\mathbb{N}\rightarrow\mathbb{N}$ is defined by

[TABLE]

where $(R/I)_{k}$ is the $k$-degree component of the gradation of $R/I$ (see [13, Section 2.2]), while the Hilbert-Poincaré series of $R/I$ is

[TABLE]

By the Hilbert-Serre theorem, the Hilbert-Poincaré series of $R/I$ is a rational function, in particular

[TABLE]

for some $h(t)\in\mathbb{Z}(t)$. There exists a unique polynomial $P_{R/I}$ such that $\operatorname{H}_{R/I}(k)=P_{R/I}(k)$ for all $k\gg 0$. The minimum integer $k_{0}\in{\mathbb{N}}$ such that $\operatorname{H}_{R/I}(k)=P_{R/I}(k)$ for all $k\geq k_{0}$ is called regularity index and we denote it by $\mathrm{ri}(R/I)$.

We end this section with the following

Remark 1.2**.**

Let $R/I_{\Delta}$ be a Stanley-Reisner ring. Then

[TABLE]

Related to the regularity index is the $a$-invariant (see Chapter 5 of [14]), namely the degree of $\mathrm{HP}_{R/I}(t)$ as a rational function, that gives further information about other algebraic invariants.

2. Reduced Euler characteristic of the independence complex

of some circulants

The goal of this section is to study the reduced Euler characteristic, $\widetilde{\chi}$, of the independence complex $\Delta(G)$ of circulant graphs, by proving bounds on the maximum clique number $\omega(\bar{G})$. In [10] the author proves that $\widetilde{\chi}(\Delta(G))\neq 0$ when $n$ is a prime number. We generalize the result for $n=p^{k}$ for any prime $p$, and $n=2p^{k}$ for any odd prime $p$. For the sake of completeness, we give the following Lemma that has been stated in [10, Lemma 1].

Lemma 2.1**.**

Let $G$ be a circulant graph on $n$ vertices. Let $f_{i-1}$ be the number of independent sets of cardinality $i$, and $f_{i-1,0}$ the number of them containing the vertex 0, then the following property holds

[TABLE]

Proof.

Let us call ${\mathcal{F}}_{i-1}\subset\Delta$ the set of faces of dimension $i-1$, that is

[TABLE]

Let $f_{i-1,j}$ be number of faces in ${\mathcal{F}}_{i-1}$ containing a given vertex $j=0,\ldots,n-1$. Since $G$ is circulant $f_{i-1,j}=f_{i-1,0}\mbox{ for all }j\in\{0,\ldots,n-1\}$. Let $A\in{\mathbb{F}}_{2}^{f_{i-1}\times n}$, $A=(a_{jk})$, be the incidence matrix, whose

[TABLE]

We observe that each row has exactly $i$ $1$-entries. Hence summing the entries of the matrix we have $if_{i-1}$. Moreover each column has exactly $f_{i-1,j}$ non-zero entries. That is $i\cdot f_{i-1}=n\cdot f_{i-1,0}$.

A useful bound on the maximum clique number for non-complete circulant graphs is given by the following

Lemma 2.2**.**

Let $G=C_{n}(S)$ be a non-complete circulant graph. Then

[TABLE]

Proof.

Suppose that $\omega(G)>\lfloor\frac{n}{2}\rfloor$. So there exists a clique $F$ of cardinality $\lfloor\frac{n}{2}\rfloor+1$. Let $r\in\{1,2,\ldots,\lfloor\frac{n}{2}\rfloor\}$. For all $v\in F$, the set $\{v+r\ :\ v\in F\}$ contains $\lfloor\frac{n}{2}\rfloor+1$ vertices so at least one of them belongs to $F$. Hence there exist $v,w\in F$, such that $w=v+r$. Since $F$ is a clique $\{v,w=v+r\}\in E(G)$, that is $r\in S$. The latter works for any $r$, then we conclude

[TABLE]

so $G$ is complete, and this contradicts our assumption.

Thanks to the Lemma 2.2, we prove the following

Theorem 2.3**.**

Let $p$ be a prime and let $G$ be a non-empty circulant graph on $n=p^{k}$ vertices with $k>0$. Then $\tilde{\chi}(\Delta(G))\neq 0.$

Proof.

By Lemma 2.1 it follows

[TABLE]

Since the graph $G$ is not empty, its complement graph $\bar{G}$ is not complete. Hence by Lemma 2.2, we have that a maximum clique in $\bar{G}$ has cardinality $d~{}<~{}\frac{p^{k}}{2}$, namely $f_{i-1}$ is a non-zero multiple of $p$ for $1\leq i\leq d$. Therefore

[TABLE]

with $r\in\mathbb{Z}$. By the primality of $p$, $\widetilde{\chi}(\Delta(G))$ is always non-zero.

Before stating the theorem on the case $n=2p^{k}$, we prove some properties that will be helpful.

Lemma 2.4**.**

Let $n=2q$ for an odd $q>1$ and let $G=C_{n}(S)$ be a non-complete circulant graph. Then $\omega(G)<q\mbox{ if and only if }\{2,4,\ldots,q-1\}\nsubseteq S.$

Proof.

$(\Rightarrow)$. By contraposition assume $\{2,4,\ldots,q-1\}\subseteq S$. We observe that the set $\{0,2,4,\ldots,2q-2\}$ is a clique of cardinality $q$. It negates the hypotesis.

$(\Leftarrow)$. Let $r\in\{2,4,\ldots,q-1\}$ be such that $r\notin S$. Let $C$ be the set of vertices of a clique of $G$. We claim $|C|<q$. We partition the set of vertices $V(G)=\{0,1,2,\ldots,2q-1\}$ in the two sets

[TABLE]

We observe that $|V_{1}|=|V_{2}|=q$.

Claim: $V_{1}$ (respectively $V_{2}$) contains at most $\frac{q-1}{2}$ vertices such that for each pair $v,w\in V_{1}$ we have $|v-w|_{n}\neq r$.

Proof of the Claim. By contraposition, assume that we take a subset $V^{\prime}$ of cardinality $\frac{q-1}{2}+1$ of $V_{1}$ with the desired property, say

[TABLE]

and since $V^{\prime}\subset V_{1}$, these are all even vertices. Now we take the set $V^{\prime\prime}=\{v+r:v\in V^{\prime}\}$. Since $r$ is even, $V^{\prime\prime}\subset V_{1}$ and $|V^{\prime}|=|V^{\prime\prime}|=\frac{q-1}{2}+1$. Since $|V^{\prime}|+|V^{\prime\prime}|>q$, then $V^{\prime}\cap V^{\prime\prime}\neq\varnothing$, so there exist $v,w\in V^{\prime}$ such that $w=v+r$; hence, the set $V^{\prime}$ has not the desired property. The claim follows.

Thus $|C\cap V_{1}|\leq\frac{q-1}{2}$ and $|C\cap V_{2}|\leq\frac{q-1}{2}$, so that $|C|<q$.

We give a generalization of Lemma 2.4 in the following

Lemma 2.5**.**

Let $n=rq$ for an odd $q>1$. Let $G=C_{n}(S)$ be a non-complete circulant graph. Then:

If \Big{\{}r,2r,\ldots,\frac{q-1}{2}r\Big{\}}\nsubseteq S then $\omega(G)\leq\frac{n-r}{2}$. 2. 2)

If $\omega(G)<q$ then \Big{\{}r,2r,\ldots,\frac{q-1}{2}r\Big{\}}\nsubseteq S.

Proof.

$1)$ The proof follows the steps of ($\Leftarrow$) of Lemma 2.4. We assume $jr\notin S$ for some $j$, $1\leq j\leq\frac{q-1}{2}$. In this case we consider the partitions $V_{i}$ of $V(G)$

[TABLE]

with $i=0,\ldots,r-1$. Let $C$ be the set of vertices of a clique of $G$. By using similar arguments to the Claim inside the proof of Lemma 2.4, we can choose at most $\frac{q-1}{2}$ vertices within each $V_{i}$ such that for each pair $v,w\in V_{i}$ we have $|v-w|_{n}\neq jr$. Hence for any $i\in\{0,\ldots,r-1\}$, it follows that $|C\cap V_{i}|\leq\frac{q-1}{2}$. Since $r\cdot(\frac{q-1}{2})=\frac{n-r}{2}$, at the end we get $|C|\leq\frac{n-r}{2}$.

$2)$ The same proof of Lemma 2.4 ($\Rightarrow$) holds.

Remark 2.6**.**

We highlight that by plugging $r=2$ in $1)$ and $2)$ of Lemma 2.5, we obtain the two implications of Lemma 2.4. It is the unique case of $n=rq$ such that $\frac{rq-r}{2}$, the bound in $1)$, is equal to $q-1$, the bound in $2)$.

For the sake of simplicity, in Proposition 2.7 and Example 2.8 we focus our attention on the clique complex of the graph.

Proposition 2.7**.**

Let $n=2p^{k}$ for an odd prime $p$, with $k>0$, and let $G=C_{n}(S)$ be a circulant graph. If $f_{p^{k}-1}$, the number of cliques of cardinality $p^{k}$, is non-zero then

[TABLE]

In particular, if one of the following condition holds

• a)

$1\notin S$,

• b)

$1\in S$* and there exists $t\in\{1,\ldots,p^{k}\}$ with $\gcd(t,2p)=1$ such that $t\notin S$,*

then $f_{p^{k}-1}=2$.

Proof.

First suppose that the graph is complete. Since $f_{p^{k}-1}=\binom{2p^{k}}{p^{k}}$ and by Lucas’s Theorem [8], we obtain

[TABLE]

Now suppose that $G$ is not complete. By Lemmas 2.2 and 2.4 since $f_{p^{k}-1}\neq 0$, that is $\omega(G)=p^{k}$, we have that $\{2,4,\ldots,p^{k}-1\}\subseteq S$. So $f_{p^{k}-1}\geq 2$. In fact the graph contains at least the two maximal cliques

[TABLE]

We observe that each clique of cardinality $p^{k}$ different from $V_{1}$ and $V_{2}$ has non-empty intersection with $V_{1}$ and $V_{2}$.

We first study the particular cases for which $f_{p^{k}-1}=2$. Suppose $1\not\in S$. In a clique $V$ of $p^{k}$ vertices different from $V_{1}$ and $V_{2}$, we must have $p^{k}$ intervals between two consecutive vertices in $V$ containing at least 1 vertex not in $V$, except for one containing 2 vertices, otherwise $V$ could be identified with $V_{1}$ or $V_{2}$. It follows $|V(G)\setminus V|\geq p^{k}-1+2=p^{k}+1$, that yields $|V|<p^{k}$. It contradicts the assumption.

Now suppose that $1\in S$ and there exists $t$ odd and coprime with $p$, $3\leq t<p^{k}$, such that $t\notin S$. We prove $f_{p^{k}-1}=2.$ By contraposition let $V$ be a clique of cardinality $p^{k}$ different from $V_{1}$ and $V_{2}$, containing [math] and $1$. Let $V^{\prime}=\{v+t\colon\ v\in V\}$. If $V\cap V^{\prime}\neq\varnothing$ there exist $v,w\in V$ such that $w=v+t$. It is impossible. If $V\cap V^{\prime}=\varnothing$, then $V(G)=V\sqcup V^{\prime}$. Since $(t,p)=1$ and $t$ odd, then $(t,n)=1$, hence there exists an odd $a\in\mathbb{Z}_{n}$, coprime with $n$, such that $at\equiv 1\ mod\ n$. Since $0\in V$ and $t\in V^{\prime}$ by definition of $V^{\prime}$ we have that $2t\in V$. In fact if $t+t=2t\in V^{\prime}$, then $t\in V$, obtaining a contradiction. It follows that

[TABLE]

The vertex $at$ lives in $V^{\prime}$ since $a$ is odd and lives in $V$ since $at=1\in V$. It implies $t\in S$. It is false.

Hence a clique of $p^{k}$ vertices different from $V_{1}$ and $V_{2}$ cannot exist and $f_{p^{k}-1}=2$.

We assume $f_{p^{k}-1}>2$. Then by the previous observations we have

[TABLE]

Now we distinguish two cases:

$S$ is the ($p^{k}-1$)-th power cycle, namely $S=\{1,\ldots,p^{k}-1\};$ 2. 2)

$S$ is not the ($p^{k}-1$)-th power cycle.

$1)$ In this case $f_{p^{k}-1}$ is the coefficient of the degree $p^{k}$ term of the independence polynomial of the graph $C_{n}(p^{k})$. As pointed out after Definition 3.4 in [2], this polynomial is

[TABLE]

hence $f_{p^{k}-1}=2^{p^{k}}\equiv_{p}2$ by Fermat’s Little Theorem.

$2)$ If $S\neq\{1,\ldots,p^{k}-1\}$, there exists an odd multiple of $p$, $mp$ for an odd $m$ with $mp<p^{k}$, such that $mp\notin S$. Let $m=q\cdot p^{r}$ for some odd $q$ with $\gcd(q,p)=1$ and $0\leq r<k-1$. Let $V$ be a clique of cardinality $p^{k}$ different from $V_{1}$ and $V_{2}$. Let $V^{\prime}=V+mp:=\{v+mp\ :v\in V\}.$ We have $V\cap V^{\prime}=\varnothing$. Moreover if $v\in V$ then $v+mp\in V^{\prime}$ and $v+2mp\in V$. In fact if $v+2mp\in V^{\prime}$ then $mp\in S$ since $V^{\prime}$ is a clique. The latter implies that $V=V+2mp=V+2qp^{r+1}$. Since $q$ is odd and coprime with $p$, it is coprime with $n$, hence it is invertible in $\mathbb{Z}_{n}$. Therefore, there exists $h\in\mathbb{Z}_{n}$ such that $qh\equiv_{n}1$, and $2qhp^{r+1}\equiv_{n}2p^{r+1}$. Since $V=V+2qp^{r+1}$, then $V=V+2p^{r+1}$. Now we prove that if $j\in\mathbb{Z}_{n}$ is such that

[TABLE]

then $2p\ |\ j$. By contradiction assume $V=V+j$ and $2p\nmid j$. We write $j=2p^{r+1}a+b$ with $0<b<2p^{r+1}$ and $2p\nmid\ b$. Since $V=V+j$, then $V=V+b$ and we have

[TABLE]

where $o(b)$ is the order of $b$ in $(\mathbb{Z}_{n},+)$. We analyze $g=\gcd(b,2p^{r+1})$ to determine the order of $b$ in $\mathbb{Z}_{n}$. Since $2p\nmid b$ and $b<2p^{r+1}$, $g$ could be either $1$, $2$, $p^{i}$ with $1\leq i\leq r+1$.

If $g=1$, then $o(b)=n$ and $L=\mathbb{Z}_{n}$, but $|V|=\frac{n}{2}$. It is impossible.

If $g=2$, then $o(b)=p^{k}$, $L=V_{1}\subseteq V$ and $|V_{1}|=|V|$ hence $V=V_{1}$. It is a contradiction to the assumption $V\neq V_{1}$.

If $g=p^{i}$, then $V=V+p^{i}=V+qp^{r+1-i}p^{i}=V^{\prime}$. It contradicts the fact $V\cap V^{\prime}=\varnothing$.

Hence, if $2p\nmid j$ then $V\neq V+j$. Let $s$ be the minimum positive integer such that $V=V+2sp$. Since $V=V+2p^{r+1}$, it follows that $s\leq p^{r}$, $2p\leq 2sp<2p^{k}$, and

[TABLE]

are $2sp$ different cliques of $G$ having cardinality $p^{k}$. Hence $2p$ divides $(f_{p^{k}-1}-2)$ and

[TABLE]

The assertion follows.

Example 2.8**.**

We provide an example with $f_{p^{k}-1}>2$ and $S\neq\{1,2,\ldots,p^{k}~{}-~{}1\}.$ We consider the graph $G=C_{50}(\{1,2,\ldots,24\}\setminus\{5\})$, using the notation of the proof of Proposition 2.7, $V=V+2mp=V+2\cdot 1\cdot 5=V+10$. The clique complex of $G$ has $32$ cliques of cardinality $25$. We fix a vertex $v$, for simplicity [math], and we look at the sequence of vertices $0,1,\ldots,9$. Moreover, the symbol [math] denotes a vertex not in a clique, while the symbol $1$ refers to a vertex in a clique. We have that $V_{1}$ has fundamental pattern $1010101010$, while $V_{2}$ has fundamental pattern $0101010101$. Since $V=V+10$, each fundamental pattern is repeated $5$ times to cover all the vertices of the graph. For example,

[TABLE]

and it happens for all the other cliques. The complex has $30$ further cliques of three kinds, namely there are three further different patterns in the sequence of vertices in or not in a clique. The other three fundamental patterns are $1111100000,\ 1110100010$ and $1101100100$. Since the graph is circulant, for each of the last three patterns, there are $10$ different cliques corresponding to the $10$ possible shifts. For example, for the first sequence we will have

[TABLE]

So the total number of cliques will be $3\cdot 10+2=32$.

Now we are able to prove

Theorem 2.9**.**

Let $p$ be an odd prime and let $G$ be a non-empty circulant graph on $n=2p^{k}$ vertices with $k>0$. Then $\widetilde{\chi}(\Delta(G))\neq 0$.

Proof.

By using similar arguments to Theorem 2.3 we say that

[TABLE]

So we write

[TABLE]

Since by Proposition 2.7 $f_{p^{k}-1}$ is [math] or it is congruent $2$ modulo $p$, we have

[TABLE]

for some $r$ in $\mathbb{Z}$. That is $\widetilde{\chi}(\Delta(G))$ does not vanish.

Example 2.10**.**

In the proofs of Theorems 2.3 and 2.9 we are giving a partial positive answer to the Conjecture 5.38 of [7] stating that all non-empty circulant graphs $G$ have $\widetilde{\chi}(\Delta(G))\neq 0$. But with a MAGMA algorithm, available at

http://www.giancarlorinaldo.it/eulercirculants.html*,*

we have found for $n=30$ and $n=36$ a list of circulants, up to isomorphisms, that contradict the Conjecture (see Table 1). Among those, for example, we report the circulant $G=C_{30}(\{1,3,8\})$ whose independence complex has $f$-vector equals to $[1,30,345,1990,6360,11736,12600,7680,2430,300]$. That is

[TABLE]

We now give some applications. The structure and roots of the independence polynomial have been studied by Hoshino and Brown (see [1], [7], [2]). By Theorem 2.3 and Theorem 2.9, we obtain the following

Corollary 2.11**.**

Let $n\in\{p^{k},2p^{k}\}$ for a prime $p$ and for $k>0$, and let $G$ be a non-empty circulant graph on $n$ vertices. Then

[TABLE]

By Example 2.10 and Equation (1.2), $-1$ is a root of the independence polynomial of the circulant graph $C_{30}(1,3,8)$.

Similar results follow by Remark 1.2 for the regularity index and the $a$-invariant. Moreover by using Corollary 4.8 of [6], we get the following

Corollary 2.12**.**

Let $G$ be a circulant graph as in Theorem 2.3 and Theorem 2.9. If $G$ is Cohen-Macaulay then

[TABLE]

It is of interest to find the hypothesis, also of different type, for the non-vanishing of the reduced Euler characteristic of circulant graphs.

We focused on the property of $n$, but nice combinatorial properties like well-coveredness (see [7], [5]), strongly connectedness (see [10]), vertex decomposability and shellability (see [12]) could be helpful. From another point of view, it would be nice to find entire classes of circulants that for particular $n$ and $S$ have vanishing Euler characteristic, by using a theoretical approach rather than the computational one used in Example 2.10.