Rocky Mountain Journal of Mathematics

Regularity of powers of edge ideals of unicyclic graphs

Ali Alilooee, Selvi Kara Beyarslan, and S. Selvaraja

Full-text: Access denied (no subscription detected)

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text


Let $G$ be a finite simple graph and $I(G)$ denote the corresponding edge ideal. In this paper, we prove that, if $G$ is a unicyclic graph, then, for all $s \geq 1$, the regularity of $I(G)^s$ is exactly $2s+\DeclareMathOperator{reg} (I(G))-2$. We also give a combinatorial characterization of unicyclic graphs with regularity $\nu (G)+1$ and $\nu (G)+2$, where $\nu (G)$ denotes the induced matching number of $G$.

Article information

Rocky Mountain J. Math., Volume 49, Number 3 (2019), 699-728.

First available in Project Euclid: 23 July 2019

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 05C25: Graphs and abstract algebra (groups, rings, fields, etc.) [See also 20F65] 05C38: Paths and cycles [See also 90B10] 05E40: Combinatorial aspects of commutative algebra 13D02: Syzygies, resolutions, complexes 13F20: Polynomial rings and ideals; rings of integer-valued polynomials [See also 11C08, 13B25]

Regularity edge ideal unicyclic graph asymptotic linearity of regularity monomial ideal


Alilooee, Ali; Beyarslan, Selvi Kara; Selvaraja, S. Regularity of powers of edge ideals of unicyclic graphs. Rocky Mountain J. Math. 49 (2019), no. 3, 699--728. doi:10.1216/RMJ-2019-49-3-699.

Export citation


  • A. Alilooee and A. Banerjee, Powers of edge ideals of regularity three bipartite graphs, J. Commutative Algebra 9 (2017), 441–454.
  • A. Banerjee, The regularity of powers of edge ideals, J. Alg. Combin. 41 (2015), 303–321.
  • A. Banerjee, S. Beyarslan, and H.T. Hà, Regularity of edge ideals and their powers, arXiv:1712.00887v2, 2017.
  • D. Berlekamp, Regularity defect stabilization of powers of an ideal, Math. Res. Lett. 19 (2012), 109–119.
  • S. Beyarslan, H.T. Hà and T.N. Trung, Regularity of powers of forests and cycles, J. Alg. Combin. 42 (2015), 1077–1095.
  • T. B\iy\ikoğlu and Y. Civan, Bounding Castelnuovo-Mumford regularity of graphs via Lozin's transformation, ArXiv e-prints 1302.3064v1, 2013.
  • T. B\iy\ikoğlu and Y. Civan, Castelnuovo-Mumford regularity of graphs, ArXiv e-prints 1503.06018, 2015.
  • M. Chardin, Some results and questions on Castelnuovo-Mumford regularity in Syzygies and Hilbert functions, Lect. Notes Pure Appl. Math. 254 (2007), 1–40.
  • ––––, Powers of ideals and the cohomology of stalks and fibers of morphisms, Alg. Num. Th. 7 (2013), 1–18.
  • A. Conca, Regularity jumps for powers of ideals, in Commutative algebra, Lect. Notes Pure Appl. Math. 244 (2006), 21–32.
  • S.D. Cutkosky, J. Herzog and N.V. Trung, Asymptotic behaviour of the Castelnuovo-Mumford regularity, Compos. Math. 118 (1999), 243–261.
  • H. Dao, C. Huneke and J. Schweig, Bounds on the regularity and projective dimension of ideals associated to graphs, J. Alg. Combin. 38 (2013), 37–55.
  • D. Eisenbud and J. Harris, Powers of ideals and fibers of morphisms, Math. Res. Lett. 17 (2010), 267–278.
  • D. Eisenbud and B. Ulrich, Notes on regularity stabilization, Proc. Amer. Math. Soc. 140 (2012), 1221–1232.
  • S. Faridi, Monomial ideals via square-free monomial ideals, in Commutative algebra, Lect. Notes Pure Appl. Math. 244 (2006), 85–114.
  • R. Fr öberg, On Stanley-Reisner rings in Topics in algebra, Part $2$, 26 Banach Center Publ. (1990), 57–70.
  • D.R. Grayson and M.E. Stillman, Macaulay2, A software system for research in algebraic geometry, available at
  • H. Hà, Asymptotic linearity of regularity and $a^\ast$-invariant of powers of ideals, Math. Res. Lett. 18 (2011), 1–9.
  • ––––, Regularity of squarefree monomial ideals in Connections between algebra, combinatorics, and geometry, Springer Proc. Math. Stat. 76 (2014), 251–276.
  • H.T. Hà, N.V. Trung, and T.N. Trung, Depth and regularity of powers of sums of ideals, Math. Z. 282 (2016), 819–838.
  • H.T. Hà and A. Van Tuyl, Monomial ideals, edge ideals of hypergraphs, and their graded Betti numbers, J. Alg. Combin. 27 (2008), 215–245.
  • J. Herzog and T. Hibi, Monomial ideals, Grad. Texts Math. 260 (2011).
  • S. Jacques, Betti numbers of graph ideals, Ph.D. dissertation, University of Sheffield.v1}, 2004.
  • A.V. Jayanthan, N. Narayanan and S. Selvaraja, Regularity of powers of bipartite graphs, J. Alg. Combin. 47 (2018), 17–38.
  • M. Katzman, Characteristic-independence of Betti numbers of graph ideals, J. Combin. Th. 113 (2006), 435–454.
  • V. Kodiyalam, Asymptotic behaviour of Castelnuovo-Mumford regularity, Proc. Amer. Math. Soc. 128 (2000), 407–411.
  • M. Moghimian, S.A. Fakhari and S. Yassemi, Regularity of powers of edge ideal of whiskered cycles, Comm. Alg. 45 (2017), 1246–1259.
  • S. Morey, Depths of powers of the edge ideal of a tree, Comm. Alg. 38 (2010), 4042–4055.
  • H.D. Nguyen and T. Vu, Powers of sums and their homological invariants, arXiv:1607.07380v3, 2016.
  • N.V. Trung and H.-Ju Wang, On the asymptotic linearity of Castelnuovo-Mumford regularity, J. Pure Appl. Alg. 201 (2005), 42–48.
  • T.N. Trung, Stability of depths of powers of edge ideals, J. Algebra 452 (2016), 157–187.
  • R. Woodroofe, Matchings, coverings, and Castelnuovo-Mumford regularity, J. Commutative Algebra 6 (2014), 287–304.
  • X. Zheng, Resolutions of facet ideals, Comm. Algebra 32 (2004), 2301–2324.