# The Regularity of Some Families of Circulant Graphs

^{1}

^{2}

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## Abstract

**:**

## 1. Introduction

## 2. Background

#### 2.1. Graph Theory Preliminaries

#### 2.2. Algebraic Preliminaries

**Theorem**

**1.**

- (i)
- if $G=H\cup K$, with H and K disjoint, then:$$\mathrm{reg}(R/I(G))=\mathrm{reg}(R/I(H))+\mathrm{reg}(R/I(K)).$$
- (ii)
- $\mathrm{reg}(I(G))=2$ if and only if ${G}^{c}$ is a chordal graph.
- (iii)
- $\mathrm{reg}(I(G))\le \mathrm{co}\text{-}\mathrm{chord}(G)+1$.
- (iv)
- if G is gap-free and claw-free, then $\mathrm{reg}(I(G))\le 3$.
- (v)
- if $x\in V(G)$, then $\mathrm{reg}(I(G))\in \{\mathrm{reg}(I(G\setminus {N}_{G}\left[x\right]))+1,\mathrm{reg}(I(G\setminus x))\}.$

**Proof.**

**Theorem 2.**

- (i)
- $\mathrm{reg}(R/I(G))\le \mathrm{reg}(R/I(H))+\mathrm{reg}(R/I(K))$, and
- (ii)
- $\mathrm{pd}(I(G))\le \mathrm{pd}(I(H))+\mathrm{pd}(I(K))+1$.

**Theorem 3.**

**Theorem**

**4.**

- (i)
- Suppose that $\mathrm{reg}(I)\le r$ and $\mathrm{pd}(I)\le n-r+1$.
- (a)
- If r is even and $\tilde{\chi}(\Delta )>0$, then $\mathrm{reg}(I)=r$.
- (b)
- If r is odd and $\tilde{\chi}(\Delta )<0$, then $\mathrm{reg}(I)=r$.

- (ii)
- Suppose that $\mathrm{reg}(I)\le r$ and $\mathrm{pd}(I)\le n-r$. If $\tilde{\chi}(\Delta )\ne 0$, then $\mathrm{reg}(I)=r$.

**Proof.**

- (i)
- If $\mathrm{reg}(I)\le r$ and $\mathrm{pd}(I)\le n-r+1$, we have ${\beta}_{a,n}(I)=0$ for all $a\le n-r-1$ and ${\beta}_{a,n}(I)=0$ for all $a\ge n-r+2$. Consequently, among all the graded Betti numbers of the form ${\beta}_{a,n}(I)$ as a varies, only ${\beta}_{n-r,n}(I)={dim}_{k}{\tilde{H}}_{r-2}(\Delta ;k)$ and ${\beta}_{n-r+1,n}(I)={dim}_{k}{\tilde{H}}_{r-3}(\Delta ;k)$ may be non-zero. Thus, by (1):$$\begin{array}{ccc}\hfill \tilde{\chi}(\Delta )& =& {(-1)}^{r-2}{dim}_{k}{\tilde{H}}_{r-2}(\Delta ;k)+{(-1)}^{r-3}{dim}_{k}{\tilde{H}}_{r-3}(\Delta ;k).\hfill \end{array}$$If we now suppose that r is even and $\tilde{\chi}(\Delta )>0$, the above expression implies:$${dim}_{k}{\tilde{H}}_{r-2}(\Delta ;k)-{dim}_{k}{\tilde{H}}_{r-3}(\Delta ;k)>0,$$
- (ii)
- Similar to Part $(i)$, the hypotheses on the regularity and projective dimension imply that $\tilde{\chi}(\Delta )={(-1)}^{r-2}{dim}_{k}{\tilde{H}}_{r-2}(\Delta ;k)={(-1)}^{r-2}{\beta}_{n-r,n}(I)$. Therefore, if $\tilde{\chi}(\Delta )\ne 0$, then ${\beta}_{n-r,n}(I)\ne 0$, which implies $\mathrm{reg}(I)=r$.

**Remark**

**1.**

## 3. The Regularity of the Edge Ideals of ${\mathit{C}}_{\mathit{n}}(\mathbf{1},\dots ,\widehat{\mathit{j}},\dots ,\lfloor \frac{\mathit{n}}{\mathbf{2}}\rfloor )$

**Lemma**

**1.**

**Proof.**

**Lemma**

**2.**

- (i)
- If $\frac{n}{d}\ge 4$, then G is claw-free.
- (ii)
- If $\frac{n}{d}\ge 5$, then G is gap free.

**Proof.**

**Theorem**

**5.**

**Proof.**

## 4. Cubic Circulant Graphs

**Theorem**

**6.**

- (a)
- If $\frac{2n}{t}$ is even, then ${C}_{2n}(a,n)$ is isomorphic to t copies of ${C}_{\frac{2n}{t}}(1,\frac{n}{t})$.
- (b)
- If $\frac{2n}{t}$ is odd, then ${C}_{2n}(a,n)$ is isomorphic to $\frac{t}{2}$ copies of ${C}_{\frac{4n}{t}}(2,\frac{2n}{t})$.

- (i)
- The family ${A}_{t}$:
- (ii)
- The family ${B}_{t}$:
- (iii)
- The family ${D}_{t}$:

**Lemma 3.**

- (i)
- If $G={A}_{t}$, then:$$\mathrm{reg}(I(G))\le \left\{\begin{array}{cc}\frac{t+4}{2}\hfill & ifteven\hfill \\ \frac{t+3}{2}\hfill & iftodd\hfill \end{array}\right.\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}and\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\mathrm{pd}(I(G))\le \left\{\begin{array}{cc}\frac{3t}{2}+1\hfill & ifteven\hfill \\ \frac{3(t-1)}{2}+2\hfill & iftodd.\hfill \end{array}\right.$$
- (ii)
- If $G={B}_{t}$, then$$\mathrm{reg}(I(G))\le \left\{\begin{array}{cc}\frac{t+4}{2}\hfill & ifteven\hfill \\ \frac{t+3}{2}\hfill & iftodd.\hfill \end{array}\right.$$
- (iii)
- If $G={D}_{t}$ and $t=2l+1$ with l an odd number, then $\mathrm{reg}(I(G))\le \frac{t+3}{2}$.

**Proof.**

**Remark 2.**

**Lemma 4.**

- (i)
- If $G={C}_{2n}(1,n)$, then:$$\mathrm{pd}(I(G))\le \left\{\begin{array}{cc}3k-1\hfill & ifn=2k\hfill \\ 3k+1\hfill & ifn=2k+1.\hfill \end{array}\right.$$
- (ii)
- If $G={C}_{2n}(2,n)$, then $\mathrm{pd}(I(G))\le 3k+1$ where $n=2k+1$.

**Proof.**

_{1}and ${A}_{2k-2}$, and the proof runs as in (i). ☐

**Lemma 5.**

- (i)
- If $G={C}_{2n}(1,n)$, then:$$\mathrm{reg}(I(G))\le \left\{\begin{array}{cc}k+1\hfill & ifn=2k,orifn=2k+1andkodd\hfill \\ k+2\hfill & ifn=2k+1andkeven.\hfill \end{array}\right.$$
- (ii)
- If $G={C}_{2n}(2,n)$, then$$\mathrm{reg}(I(G))\le \left\{\begin{array}{cc}k+1\hfill & ifn=2k+1andkeven\hfill \\ k+2\hfill & ifn=2k+1andkodd.\hfill \end{array}\right.$$

**Proof.**

_{1}, i.e.,

**Theorem 7**

**.**For each $n\ge 3$, set:

- (i)
- If $G={C}_{2n}(1,n)$ with n even, or if $G={C}_{2n}(2,n)$ with n odd, then $I(G,x)={I}_{n}(x)$.
- (ii)
- If $G={C}_{2n}(1,n)$ and n is odd, then $I(G,x)={I}_{n}(x)+2{x}^{n}$.

**Theorem 8.**

- (a)
- If $\frac{2n}{t}$ is even, then:$$\mathrm{reg}(I({C}_{2n}(a,n)))=\left\{\begin{array}{cc}kt+1\hfill & if\frac{n}{t}=2k,or\frac{n}{t}=2k+1withkanoddnumber\hfill \\ (k+1)t+1\hfill & if\frac{n}{t}=2k+1withkanevennumber.\hfill \end{array}\right.$$
- (b)
- If $\frac{2n}{t}$ is odd, then:$$\mathrm{reg}(I({C}_{2n}(a,n)))=\left\{\begin{array}{cc}\frac{kt}{2}+1\hfill & if\frac{2n}{t}=2k+1withkanevennumber\hfill \\ \frac{(k+1)t}{2}+1\hfill & if\frac{2n}{t}=2k+1withkanoddnumber.\hfill \end{array}\right.$$

**Proof.**

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

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Uribe-Paczka, M.E.; Van Tuyl, A. The Regularity of Some Families of Circulant Graphs. *Mathematics* **2019**, *7*, 657.
https://doi.org/10.3390/math7070657

**AMA Style**

Uribe-Paczka ME, Van Tuyl A. The Regularity of Some Families of Circulant Graphs. *Mathematics*. 2019; 7(7):657.
https://doi.org/10.3390/math7070657

**Chicago/Turabian Style**

Uribe-Paczka, Miguel Eduardo, and Adam Van Tuyl. 2019. "The Regularity of Some Families of Circulant Graphs" *Mathematics* 7, no. 7: 657.
https://doi.org/10.3390/math7070657