Homological invariants of edge ideals of weighted oriented graphs

TL;DR

This work extends the theory of edge ideals to weighted oriented graphs by characterizing all possible triples of depth, dimension, and regularity for connected weighted oriented graphs on a fixed number of vertices, and by determining the full range of Betti-table sizes for edge ideals of weighted oriented trees and connected bipartite graphs. It leverages depth-zero criteria via dominant monomials, Betti-splitting techniques, and a construction toolkit to relate weighted and unweighted cases, yielding explicit descriptions of the sets $ ext{DD}(n)$, $ ext{DDR}(n)$, and Betti-table-size sets $ ext{PR}^{ ext{BPT}}(n)$ and $ ext{PR}^{ ext{TREE}}(n)$ for $n\ge4$. The results generalize the unweighted graph tuples studied by Kanno Kannos-tuples, and provide concrete, computable invariants for a broad class of monomial ideals arising from weighted oriented graphs. Overall, the paper advances understanding of how weights and orientations influence homological invariants and Betti tables in edge ideals, with implications for coding theory and combinatorial commutative algebra.

Abstract

We determine all possible triples of depth, dimension, and regularity of edge ideals of weighted oriented graphs with a fixed number of vertices. Also, we compute all the possible Betti table sizes of edge ideals of weighted oriented trees and bipartite~graphs with a fixed number of vertices.

Figures (7)

• Figure 1: Examples of a unicyclic graph (left) and a bipartite graph (right).
• Figure 2: A tree (left) and the same tree as rooted at vertex $x$ (right).
• Figure 3: The decomposition of the unicyclic graph in \ref{['fig:ex-unicyclic-bipartite']} into a cycle and two trees.
• Figure 4: A naturally oriented unicyclic graph (above) and two "unnatural" orientations of the same graph (below).
• Figure 5: A weighted oriented graph (left) and its underlying base graph (right).

Theorems & Definitions (43)

• Theorem 1.1: \ref{['thm:dd-wo']} and \ref{['thm:ddr-wo']}
• Theorem 1.2: \ref{['thm:bipartite-wo']} and \ref{['thm:tree-wo']}
• Definition 2.1
• Proposition 2.2
• proof
• Lemma 2.3
• proof
• Example 2.4
• Lemma 2.5: HH11-book
• Proposition 2.6