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arXiv · 2205.07059

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

Chwas Ahmed·Amir Mafi·Mohammed Rafiq Namiq
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Abstract

We introduce a class of chordal graphs called ($d_1$,$d_2$,$\dots$,$d_q$)-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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Chwas Ahmed, Amir Mafi, Mohammed Rafiq Namiq. 2022-05-14. Sequentially Cohen-Macaulay Co-Chordal Graphs: Structure and Projective Dimension. https://arxiv.org/abs/2205.07059

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