2017/05/29 by S. A. Seyed Fakhari, Siamak Yassemi, Fakhari, Seyed Amin Seyed +1
Computer Science · Mathematics · #Combinatorics (math.CO) #Commutative Algebra (math.AC) #Commutative Algebra and Its Applications #FOS: Mathematics #Polynomial and algebraic computation #Rings, Modules, and Algebras
paper · pdf · doi:10.48550/arxiv.1705.10226
openalex publication_date 2017/05/29 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Let G be a graph with n vertices, S=\mathbbK[x1,…,xn] be the polynomial ring in n variables over a field \mathbbK and I(G) denote the edge ideal of G. For every collection H of connected graphs with K2∈ H, we introduce the notions of \ind-matchH(G) and min-matchH(G). It will be proved that the inequalities \ind-match_\K2, C5\(G)≤\rm reg(S/I(G))≤min-match_\K2, C5\(G) are true. Moreover, we show that if G is a Cohen--Macaulay graph with girth at least five, then \rm reg(S/I(G))=\ind-match_\K2, C5\(G). Furthermore, we prove that if G is a paw--free and doubly Cohen--Macaulay graph, then \rm reg(S/I(G))=\ind-match_\K2, C5\(G) if and only if every connected component of G is either a complete graph or a 5-cycle graph. Among other results, we show that for every doubly Cohen--Macaulay simplicial complex, the equality \rm reg(\mathbbK[Δ])=\rm dim(\mathbbK[Δ]) holds.