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Abstract In this paper, we investigate the degree ofh-polynomials of edge ideals of finite simple graphs. In particular, we provide combinatorial formulas for the degree of theh-polynomial for various fundamental classes of graphs such as paths, cycles, and bipartite graphs. To the best of our knowledge, this study represents the first investigation into the combinatorial interpretation of this algebraic invariant. Additionally, we characterize all connected graphs in which the sum of the Castelnuovo–Mumford regularity and the degree of theh-polynomial of an edge ideal achieve its maximum value, equal to the number of vertices in the graph.
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Enumerating combinatorial resultant trees
A 2D rigidity circuit is a minimal graph G = (V , E) supporting a non-trivial stress in any generic placement of its vertices in the Euclidean plane. All 2D rigidity circuits can be constructed from K4 graphs using combinatorial resultant (CR) operations. A combinatorial resultant tree (CR-tree) is a rooted binary tree capturing the structure of such a construction. The CR operation has a specific algebraic interpretation, where an essentially unique circuit polynomial is associated to each circuit graph. Performing Sylvester resultant operations on these polynomials is in one-to-one correspondence with CR operations on circuit graphs. This mixed combinatorial/algebraic approach led recently to an effective algorithm for computing circuit polynomials. Its complexity analysis remains an open problem, but it is known to be influenced by the depth and shape of CR-trees in ways that have only partially been investigated. In this paper, we present an effective algorithm for enumerating all the CR-trees of a given circuit with n vertices. Our algorithm has been fully implemented in Mathematica and allows for computational experimentation with various optimality criteria in the resulting, potentially exponentially large collections of CR-trees.
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- Award ID(s):
- 2212309
- PAR ID:
- 10511459
- Publisher / Repository:
- Elsevier Science Direct
- Date Published:
- Journal Name:
- Computational Geometry
- Volume:
- 118
- Issue:
- C
- ISSN:
- 0925-7721
- Page Range / eLocation ID:
- 102064
- Subject(s) / Keyword(s):
- Rigidity matroid Circuit polynomial Combinatorial resultant Inductive construction Resultant tree
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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- Free Publicly Accessible Full Text
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Accepted Manuscript1.0
- Journal Article:
- https://doi.org/10.1016/j.comgeo.2023.102064
