A note on the algebra of threshold graphs
DOI:
https://doi.org/10.13069/jacodesmath.v13i2.396Keywords:
Graphs, Edge rings, Neighborhood complexAbstract
It is known that threshold graphs have edge rings with $2-$linear resolutions. This was proved by Engström and Stamps, [4]. They used the fact that an edge ring of a graph $G$ has a $2-$linear resolution if and only if the complement graph is chordal. They also described a method to determine the Betti numbers. Our goal is to determine when edge rings of threshold graphs are Cohen-Macaulay. In order to do so, it is more convenient to use an alternative way to study edge rings of graphs, that is to interpret them as Stanley-Reisner rings. We also determine when the neighborhood complex of a threshold graph has a Cohen-Macaulay Stanley-Reisner ring.
Accepted: 28 October 2025
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References
V. Chvátal and P. L. Hammer, Aggregation of inequalities in integer programming in: P. L. Hammer, E. L. Johnson, B. H. Korte, and G. L. Nemhauser, eds., Studies in Integer Programming, vol. 1 in Annals of Discrete Mathematics, North-Holland (1977) 145–162 .
A. Dochtermann and A. Engström, Algebraic properties of edge ideals via combinatorical topology, Electron. J. Combin. 16(2) (2009) #R2.
J. A. Eagon and V. Reiner, Resolutions of Stanley-Reisner rings and Alexander duality, J. Pure Appl. Algebra, 130(3) (1998) 265–275.
A. Engström and M. T. Stamps, Betti diagrams from graphs, Algebra Number Theory 7(7) (2013) 1725–1742.
R. Fröberg, On Stanley-Reisner rings, Banach Center Publ. 26(2) (1990) 57–70.
R. Fröberg, Betti numbers of fat forests and their Alexander dual, J. Alg. Comb. 56 (2022) 1023–1030.
C. J. Klivans, Threshold graphs, shifted complexes, and graphical complexes, Discrete Math. 307(21) (2007) 2591–2597.
N. V. R. Mahadev and U. N. Peled, eds., Threshold Graphs and Related Topics, vol. 56 in Annals of Discrete Mathematics, North-Holland (1995).
T. Matsushita and S. Wakatsuki, Dominance complexes, neighborhood complexes and combinatorial Alexander dual, J. Combin. Theory Ser. A 211 (2025) 105978.
N. Terai, Generalization of Eagon-Reiner theorem and h-vectors of graded rings, Preprint.
N. Terai, Alexander duality theorem and Stanley-Reisner rings, Free resolutions of coordinate rings of projective varieties and related topics, (1998), Sūrikaisekikenkyūsho Kōkyūroku 1078 (1999) 174–184.
