Sequentially Cohen-Macaulay Co-Chordal Graphs: Structure and Projective Dimension
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arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2022
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| _version_ | 1866909835617894400 |
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| author | Ahmed, Chwas Mafi, Amir Namiq, Mohammed Rafiq |
| author_facet | Ahmed, Chwas Mafi, Amir Namiq, Mohammed Rafiq |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2205_07059 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Sequentially Cohen-Macaulay Co-Chordal Graphs: Structure and Projective Dimension Ahmed, Chwas Mafi, Amir Namiq, Mohammed Rafiq Commutative Algebra Combinatorics Primary 05C75, 13D02, Secondary 05E40, 13F55, 05C69 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. |
| title | Sequentially Cohen-Macaulay Co-Chordal Graphs: Structure and Projective Dimension |
| topic | Commutative Algebra Combinatorics Primary 05C75, 13D02, Secondary 05E40, 13F55, 05C69 |
| url | https://arxiv.org/abs/2205.07059 |