Abstract
Let D be a weighted oriented graph with the underlying graph G and I(D), I(G) be the edge ideals corresponding to D and G respectively. We show that the regularity of edge ideal of a certain class of weighted oriented graph remains same even after adding certain kind of new edges to it. We also establish the relationship between the regularity of edge ideal of weighted oriented path and cycle with the regularity of edge ideal of their underlying graph when vertices of \(V^+\) are sinks.
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References
A. Alilooee and S. Faridi, On the resolution of path ideals of cycles, Comm. Algebra 43 (2015), no. 12, 5413–5433.
A. Alilooee and S. Faridi, Graded Betti numbers of path ideals of cycles and lines, J. Algebra Appl. 17 (2018), no. 1, 1850011-1-17.
S. Beyarslan, T. Huy, T. Trung and N. Nam, Regularity of powers of forests and cycles, J. Algebraic Combin. 42 (2015), no. 4, 1077–1095.
S. K. Beyarslan, J. Biermann, K-N. Lin and A. O’Keefe, Algebraic invariants of weighted oriented graphs, ar**v:1910.11773.
R. R. Bouchat, H. T. Hà and A. O’Keefe, Path ideals of rooted trees and their graded Betti numbers, J. Comb. Theory Ser. A 118 (2011), no. 8, 2411–2425.
S. Eliahou and M. Kervaire, Minimal resolutions of some monomial ideals, J. Algebra 129 (1990), no. 1, 1-25.
G. Fatabbi, On the resolution of ideals of fat points, J. Algebra 242 (2001), no. 1, 92-108.
C. A. Francisco, H. T. Hà and A. Van Tuyl, Splittings of monomial ideals, Proc. Amer. Math. Soc. 137 (2009), no. 10, 3271–3282.
P. Gimenez, J. M. Bernal, A. Simis, R. H. Villarreal and C. E. Vivares, Symbolic powers of monomial ideals and Cohen-Macaulay vertex-weighted digraphs, Singularities, algebraic geometry, commutative algebra, and related topics, Springer, Cham, 2018, 491-510.
H. T. Hà, N. V. Trung and T. N. Trung, Depth and regularity of powers of sums of ideals, Math. Z. 282 (3-4) (2016), 819-838.
H. T. Hà, K-N Lin, S. Morey, E. Reyes and R. H. Villarreal, Edge ideals of oriented graphs, Internat. J. Algebra Comput. 29 (2019), no. 3, 535-559.
H.T. Hà, Regularity of squarefree monomial ideals, Connections Between Algebra, Combinatorics and Geometry, Springer Proc. Math. Stat. 76 (2014), 251-276.
J. Herzog and T. Hibi, Monomial Ideals, New York, NY, USA: Springer-Verlag, 2011.
S. Jacques, Betti numbers of graph ideals, PhD dissertation, University of Sheffield, 2004.
D. Kiani and S. S. Madani, Betti numbers of path ideals of trees, Comm. Algebra 44 (2016), no. 12, 5376–5394.
K-N. Lin and J. McCullough, Hypergraphs and regularity of square-free monomial ideals, Internat. J. Algebra Comput. 23 (2013), no. 7, 1573-1590.
J. Martínez-Bernal, Y. Pitones and R. H. Villarreal, Minimum distance functions of graded ideals and Reed-Muller-type codes, J. Pure Appl. Algebra 221 (2017), no. 2, 251-275.
Y. Pitones, E. Reyes and J. Toledo, Monomial ideals of weighted oriented graphs, Electron. J. Combin. 26 (2019), no. 3, 1-18.
G. Zhu, Projective dimension and regularity of the path ideal of the line graph, J. Algebra Appl. 17 (2018), no. 4, 1850068-1-15.
G. Zhu, L. Xu, H. Wang and Z. Tang, Projective dimension and regularity of edge ideal of some weighted oriented graphs, Rocky MT J. Math. 49 (2019), no. 4, 1391-1406.
G. Zhu, L. Xu, H. Wang and Z. Tang, Projective dimension and regularity of edge ideals of some vertex-weighted oriented \( m- \)partite graphs, ar**v:1904.04682.
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Communicated by Jugal K Verma.
M. Mandal: Supported by SERB(DST) Grant No.: EMR/2016/006997, India.
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Mandal, M., Pradhan, D.K. Regularity in weighted oriented graphs. Indian J Pure Appl Math 52, 1055–1071 (2021). https://doi.org/10.1007/s13226-021-00039-2
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DOI: https://doi.org/10.1007/s13226-021-00039-2