# The Regularity of Some Families of Circulant Graphs

^{1}

^{2}

^{*}

## 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

## References

- Morey, S.; Villarreal, R.H. Edge ideals: Algebraic and combinatorial properties. In Progress in Commutative Algebra 1; de Gruyter: Berlin, Gemany, 2012; pp. 85–126. [Google Scholar]
- Villarreal, R.H. Monomial algebras. In Monographs and Research Notes in Mathematics; CRC Press: Boca Raton, FL, USA, 2015. [Google Scholar]
- Hà, H.T. Regularity of squarefree monomial ideals. In Connections between Algebra, Combinatorics, and Geometry; Springer Proceedings in Mathematics & Statistics: New York, NY, USA, 2014; Volume 76, pp. 251–276. [Google Scholar] [CrossRef]
- Earl, J.; Vander Meulen, K.N.; Van Tuyl, A. Independence complexes of well-covered circulant graphs. Experiment. Math.
**2016**, 25, 441–451. [Google Scholar] [CrossRef] - Makvand, M.A.; Mousivand, A. Betti numbers of some circulant graphs. To appear Czechoslov. Math. J.
**2019**. [Google Scholar] [CrossRef] - Mousivand, A. Circulant S
_{2}graphs. Preprint**2015**, arXiv:1512.08141. [Google Scholar] - Rinaldo, G. Some algebraic invariants of edge ideal of circulant graphs. Bull. Math. Soc. Sci. Math. Roumanie (N.S.)
**2018**, 61, 95–105. [Google Scholar] - Rinaldo, G.; Romeo, F. On the reduced Euler characteristic of independence complexes of circulant graphs. Discrete Math.
**2018**, 341, 2380–2386. [Google Scholar] [CrossRef] [Green Version] - Rinaldo, G.; Romeo, F. 2-Dimensional vertex decomposable circulant graphs. Preprint
**2018**, arXiv:1807.05755. [Google Scholar] - Romeo, F. Chordal circulant graphs and induced matching number. Preprint
**2018**, arXiv:1811.06409. [Google Scholar] - Vander Meulen, K.N.; Van Tuyl, A.; Watt, C. Cohen-Macaulay Circulant Graphs. Comm. Alg.
**2014**, 42, 1896–1910. [Google Scholar] [CrossRef] - Fröberg, R. On Stanley-Reisner rings. In Topics in Algebra, Part 2 (Warsaw, 1988); Banach Center Publ.: Warsaw, Poland, 1990; Volume 2, pp. 57–70. [Google Scholar]
- Jacques, S. Betti Numbers of Graph Ideals. Ph.D. Thesis, University of Sheffield, Sheffield, UK, 2004. [Google Scholar]
- Nevo, E. Regularity of edge ideals of C
_{4}-free graphs via the topology of the lcm-lattice. J. Combin. Theory Ser. A**2011**, 118, 491–501. [Google Scholar] [CrossRef] - Woodroofe, R. Matchings, coverings, and Castelnuovo–Mumford regularity. J. Commut. Algebra
**2014**, 6, 287–304. [Google Scholar] [CrossRef] - Dao, H.; Huneke, C.; Schweig, J. Bounds on the regularity and projective dimension of ideals associated to graphs. J. Algebraic Combin.
**2013**, 38, 37–55. [Google Scholar] [CrossRef] - Kalai, G.; Meshulam, R. Intersections of Leray complexes and regularity of monomial ideals. J. Combin. Theory Ser. A
**2006**, 113, 1586–1592. [Google Scholar] [CrossRef] [Green Version] - Davis, G.J.; Domke, G.S. 3-Circulant Graphs. J. Combin. Math. Combin. Comput.
**2002**, 40, 133–142. [Google Scholar] - Grayson, D.; Stillman, M. Macaulay 2, a Software System for Research in Algebraic Geometry. Available online: http://www.math.uiuc.edu/Macaulay2/ (accessed on 20 July 2019).
- Katzman, M. Characteristic-independence of Betti numbers of graph ideals. J. Combin. Theory Ser. A
**2006**, 113, 435–454. [Google Scholar] [CrossRef] [Green Version] - Hoshino, R. Independence Polynomials of Circulant Graphs. Ph.D. Thesis, Dalhouise University, Halifax, NS, Canada, 2008. [Google Scholar]
- Brown, J.; Hoshino, R. Well-covered circulant graphs. Discrete Math.
**2011**, 311, 244–251. [Google Scholar] [CrossRef]

© 2019 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

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