Toric Rings of Weighted Oriented Graphs

Given a vertex-weighted oriented graph, we can associate to it a set of monomials. We consider the toric ideal whose defining map is given by these monomials. We find a generating set for the toric ideal for certain classes of graphs which depends on the combinatorial structure and weights of the graph. We provide a result analogous to the unweighted, unoriented graph case, to show that when the associated simple graph has only trivial even closed walks, the toric ideal is the zero ideal. Moreover, we give necessary and sufficient conditions for the toric ideal of a weighted oriented graph to be generated by a single binomial and we describe the binomial in terms of the structure of the graph.

Electronic version of an article published as International Journal of Algebra and Computation, 32,(2), 2022, 327-345, doi: 10.1142/s0218196722500151, © World Scientific Publishing Company, https://doi.org/10.1142/S0218196722500151

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Work Title Toric Rings of Weighted Oriented Graphs
Access
Open Access
Creators
  1. Jennifer Biermann
  2. Selvi Kara
  3. Kuei-Nuan Lin
  4. Augustine O’Keefe
Keyword
  1. Graph
  2. Weighted Graph
  3. Edge Ideal
  4. Toric Ideal
License CC BY 4.0 (Attribution)
Work Type Article
Publisher
  1. World Scientific Pub Co Pte Ltd
Publication Date February 11, 2022
Publisher Identifier (DOI)
  1. 10.1142/s0218196722500151
Source
  1. International Journal of Algebra and Computation
Deposited May 09, 2022

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Version 1
published

  • Created
  • Added ToricWeightedJul9-1.pdf
  • Added Creator Jennifer Biermann
  • Added Creator Selvi Kara
  • Added Creator Kuei-Nuan Lin
  • Added Creator Augustine O’Keefe
  • Published
  • Updated Work Title, Description Show Changes
    Work Title
    • Toric Rings of Weighted Oriented Graphs
    • ! Toric Rings of Weighted Oriented Graphs
    Description
    • <jats:p> Given a vertex-weighted oriented graph, we can associate to it a set of monomials. We consider the toric ideal whose defining map is given by these monomials. We find a generating set for the toric ideal for certain classes of graphs which depends on the combinatorial structure and weights of the graph. We provide a result analogous to the unweighted, unoriented graph case, to show that when the associated simple graph has only trivial even closed walks, the toric ideal is the zero ideal. Moreover, we give necessary and sufficient conditions for the toric ideal of a weighted oriented graph to be generated by a single binomial and we describe the binomial in terms of the structure of the graph. </jats:p>
    • Given a vertex-weighted oriented graph, we can associate to it a set of monomials. We consider the toric ideal whose defining map is given by these monomials. We find a generating set for the toric ideal for certain classes of graphs which depends on the combinatorial structure and weights of the graph. We provide a result analogous to the unweighted, unoriented graph case, to show that when the associated simple graph has only trivial even closed walks, the toric ideal is the zero ideal. Moreover, we give necessary and sufficient conditions for the toric ideal of a weighted oriented graph to be generated by a single binomial and we describe the binomial in terms of the structure of the graph.
  • Updated Keyword Show Changes
    Keyword
    • Graph, Weighted Graph, Edge Ideal, Toric Ideal
  • Updated Work Title Show Changes
    Work Title
    • ! Toric Rings of Weighted Oriented Graphs
    • Toric Rings of Weighted Oriented Graphs
  • Updated