Graded Betti numbers of a hyperedge ideal associated to join of graphs

2023/11/13 by Sourav Kanti Patra, Patra, Sourav Kanti, Amit Roy +1 · 1 citation
Computer Science · Mathematics · Medicine · #05E45 #13F55 #Cholinesterase and Neurodegenerative Diseases #Combinatorics (math.CO) #Commutative Algebra (math.AC) #Commutative Algebra and Its Applications #FOS: Mathematics #Topological and Geometric Data Analysis

paper · pdf · doi:10.48550/arxiv.2311.07308

openalex publication_date 2023/11/13 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Let G be a finite simple graph on the vertex set V(G) and let r be a positive integer. We consider the hypergraph Con_r(G) whose vertices are the vertices of G and the (hyper)edges are all A⊆ V(G) such that |A|=r+1 and the induced subgraph G[A] is connected. The (hyper)edge ideal I_r(G) of Con_r(G) is also the Stanley-Reisner ideal of a generalisation of the independence complex of G, called the r-independence complex Ind_r(G). In this article we make extensive use of the Mayer-Vietoris sequence to find the graded Betti numbers of I_r(G₁*G₂) in terms of the graded Betti numbers of I_r(G₁) and I_r(G₂), where G₁*G₂ is the join of G₁ and G₂. Moreover, we find formulas for all the graded Betti numbers of I_r(G), when G is a complete graph, complete multipartite graph, cycle graph and the wheel graph.

Cited by

• The complex of r-co-connected subgraphs, chordality and Fröberg's theorem

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