Abstract
In this paper we will provide the exact formulas for the regularity and projective dimension of edge ideals of three types of vertex-weighted oriented m-partite graphs. We also give some examples to show that the restrictions on orientations of edges and weights of vertices can not be removed.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Bondy, J.A., Murty, U.S.R.: Graph Theory. Springer, Berlin (2008)
Caviglia, G., Hà, H.T., Herzog, J., Kummini, M., Terai, N., Trung, N.V.: Depth and regularity modulo a principal ideal. J. Algebraic Comb. 49(1), 1–20 (2019)
CoCoATeam.: CoCoA: a system for doing computations in commutative algebra. http://cocoa.dima.unige.it
Francisco, C.A., Hà, H.T., Van Tuyl, A.: Splittings of monomial ideals. Proc. Am. Math. Soc. 137(10), 3271–3282 (2009)
Gimenez, P., Bernal, J.M., Simis, A., Villarreal, R.H., Vivares, C.E.: Symbolic Powers of Monomial Ideals and Cohen–Macaulay Vertex-Weighted Digraphs, Singularities, Algebraic Geometry, Commutative Algebra, and Related Topics, pp. 491–510. Springer, Cham (2018)
Hà, H.T., Van Tuyl, A.: Splittable ideals and the resolutions of monomial ideals. J. Algebra 309(1), 405–425 (2007)
Hà, H.T., Lin, K.N., Morey, S., Reyes, E., Villarreal, R.H.: Edge ideals of oriented graphs. Int. J. Algebra Comput. 29(3), 535–559 (2019)
Herzog, J., Hibi, T.: Monomial Ideals. Springer, New York (2011)
Hoa, L.T., Tam, N.D.: On some invariants of a mixed product of ideals. Arch. Math. 94(4), 327–337 (2010)
Jensen, J.B., Gutin, G.: Digraphs. Theory, Algorithms and Applications. Springer Monographs in Mathematics. Springer, Berlin (2006)
Martínez-Bernal, J., Pitones, Y., Villarreal, R.H.: Minimum distance functions of graded ideals and Reed–Muller-type codes. J. Pure Appl. Algebra 221, 251–275 (2017)
Paulsen, C., Sather-Wagstaff, S.: Edge ideals of weighted graphs. J. Algebra Appl. 12(5), 1250223124 (2013)
Pitones, Y., Reyes, E., Toledo, J.: Monomial ideals of weighted oriented graphs. Electron. J. Comb. 26(3), 1–18 (2019)
Villarreal, R.H.: Cohen–Macaulay graphs. Manuscr. Math. 66(3), 277–293 (1990)
Xu. L., Zhu, G., Wang, H., Zhang, J.: Projective dimension and regularity of powers of edge ideals of vertex-weighted rooted forests. Bull. Malaysian Math. Sci. Soc. (2021). https://doi.org/10.1007/s40840-020-01052-0
Zhu, G.: Projective dimension and regularity of the path ideal of the line graph. J. Algebra Appl. 17(4), 1850068115 (2018a)
Zhu, G.: Projective dimension and regularity of path ideals of cycles. J. Algebra Appl. 17(10), 1850188122 (2018b)
Zhu, G., Li, X., Wang, H., Tang, Z.: Projective dimension and regularity of edge ideals of some weighted oriented graphs. Rocky Mt. J. Math. 49(4), 1391–1406 (2019)
Zhu, G., Xu. L., Wang, H., Zhang, J.: Projective dimension and regularity of edge ideal of the disjoint union of some weighted oriented gapfree bipartite graphs. J. Algebra Appl. 19(12), 2050233-1-23 (2020)
Acknowledgements
This research is supported by the National Natural Science Foundation of China (No. 11271275) and by foundation of the Priority Academic Program Development of Jiangsu Higher Education Institutions.
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
About this article
Cite this article
Zhu, G., Wang, H., Xu, L. et al. Algebraic Properties of Edge Ideals of Some Vertex-Weighted Oriented m-Partite Graphs. Bull Braz Math Soc, New Series 52, 1005–1023 (2021). https://doi.org/10.1007/s00574-021-00242-z
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00574-021-00242-z