2023/06/08 by Antonino Ficarra, Jürgen Herzog, Ficarra, Antonino +3
Computer Science · Mathematics · #Commutative Algebra (math.AC) #Commutative Algebra and Its Applications #FOS: Mathematics #Graph theory and applications #Topological and Geometric Data Analysis
paper · pdf · doi:10.48550/arxiv.2306.05020
openalex publication_date 2023/06/08 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Let Δ be a 1-dimensional simplicial complex. Then Δ may be identified with a finite simple graph G. In this article, we investigate the toric ring RG of G. All graphs G such that RG is a normal domain are classified. For such a graph, we determine the set PG of height one monomial prime ideals of RG. In the bipartite case, and in the case of whiskered cycles, this set is explicitly described. As a consequence, we determine the canonical class [ωRG] and characterize the Gorenstein property of RG. For a bipartite graph G, we show that RG is Gorenstein if and only if G is unmixed. For a subclass of non-bipartite graphs G, which includes whiskered cycles, RG is Gorenstein if and only if G is unmixed and has an odd number of vertices. Finally, it is proved that RG is a pseudo-Gorenstein ring if G is an odd cycle.