Sequentially Cohen-Macaulay Co-Chordal Graphs: Structure and Projective Dimension

2022/05/14 by Chwas Ahmed, Амир Мафи, Ahmed, Chwas +3 · 1 citation
Mathematics · Medicine · #05C69 #13D02 #13F55 #Algebraic structures and combinatorial models #Cholinesterase and Neurodegenerative Diseases #Combinatorics (math.CO) #Commutative Algebra (math.AC) #Commutative Algebra and Its Applications #FOS: Mathematics #Primary 05C75 #Secondary 05E40

paper · pdf · doi:10.48550/arxiv.2205.07059

openalex publication_date 2022/05/14 · openalex created_date 2022/05/22 · openalex updated_date 2026/07/30

Abstract

We introduce a class of chordal graphs called (d₁,d₂,…,d_q)-trees. A graph belongs to this class if and only if its clique complex is sequentially Cohen-Macaulay, providing a complete classification of all sequentially Cohen-Macaulay co-chordal graphs. This class also yields a classification of bi-sequentially Cohen-Macaulay graphs. We study the relationship between the projective dimension of a graph and its maximum vertex degree. We show that the projective dimension is always at least the maximum vertex degree, although this bound is not always tight, even for co-chordal graphs. However, equality holds when the graph is sequentially Cohen-Macaulay co-chordal or has a full vertex.

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