The edge rings of compact graphs

2023 · doi:10.32388/ifin2w
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This paper classifies compact graphs by proving that their edge rings have Cohen-Macaulay type and projective dimension equal to the number of induced cycles minus one, and regularity equal to the matching number of a derived graph.

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⚙ AI-generated deep summary by claude@2026-07, 2026-07-15 · read from full text ⓘ

This paper studies “edge rings” associated with compact graphs, focusing on algebraic properties that arise from the combinatorial structure of such graphs. It develops definitions and investigates how the edge ring behavior relates to features of the underlying graph setting. A key finding is the characterization of properties of these edge rings in the compact-graph context. The paper’s main limitation is that the results are framed within graph-theoretic objects (compact graphs) rather than biomedical samples or experimental systems. The paper does not explicitly discuss endometriosis or adenomyosis; it was included in the corpus via a keyword match in the upstream search index.

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Abstract

We define a simple graph as compact if it lacks even cycles and satisfies the odd-cycle condition. Our focus is on classifying all compact graphs and examining the characteristics of their edge rings. Let \(G\) be a compact graph and \(\mathbb{K}[G]\) be its edge ring. Specifically, we demonstrate that the Cohen-Macaulay type and the projective dimension of \(\mathbb{K}[G]\) are both equal to the number of induced cycles of \(G\) minus one and that the regularity of \(\mathbb{K}[G]\) is equal to the matching number of \(G_{0}\). Here, \(G_{0}\) is obtained from \(G\) by removing the vertices of degree one successively, resulting in a graph where every vertex has a degree greater than \(1\).
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