A glimpse to most of the old and new results on very well-covered graphs from the viewpoint of commutative algebra

K. Kimura; M. R. Pournaki; S. A. Seyed Fakhari; N. Terai; S. Yassemi

doi:10.1007/s40687-022-00326-2

A glimpse to most of the old and new results on very well-covered graphs from the viewpoint of commutative algebra

K. Kimura, M. R. Pournaki, S. A. Seyed Fakhari, N. Terai, S. Yassemi

Research output: Contribution to journal › Article › peer-review

Abstract

A very well-covered graph is a well-covered graph without isolated vertices such that the height of its edge ideal is half of the number of vertices. In this survey article, we gather together most of the old and new results on the edge and cover ideals of these graphs.

Original language	English
Article number	29
Journal	Research in Mathematical Sciences
Volume	9
Issue number	2
DOIs	https://doi.org/10.1007/s40687-022-00326-2
Publication status	Published - Jun 2022

Keywords

Betti number
CM property
Cohen–Macaulay graph
Cohen–Macaulay ring
Compression
Cover ideal
Depth
Edge ideal
Edge-weighted ideal
f-Vector
Flag complex
h-Vector
Height
Independence complex
Linear resolution
Local cohomology
Minimal free resolution
Projective dimension
Regularity
Shellability
Simplicial complex
Stanley–Reisner ideal
Symbolic power
Vertex-decomposability
Very well-covered graph
Well-covered graph

ASJC Scopus subject areas

Theoretical Computer Science
Mathematics (miscellaneous)
Computational Mathematics
Applied Mathematics

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10.1007/s40687-022-00326-2

Cite this

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title = "A glimpse to most of the old and new results on very well-covered graphs from the viewpoint of commutative algebra",

abstract = "A very well-covered graph is a well-covered graph without isolated vertices such that the height of its edge ideal is half of the number of vertices. In this survey article, we gather together most of the old and new results on the edge and cover ideals of these graphs.",

keywords = "Betti number, CM property, Cohen–Macaulay graph, Cohen–Macaulay ring, Compression, Cover ideal, Depth, Edge ideal, Edge-weighted ideal, f-Vector, Flag complex, h-Vector, Height, Independence complex, Linear resolution, Local cohomology, Minimal free resolution, Projective dimension, Regularity, Shellability, Simplicial complex, Stanley–Reisner ideal, Symbolic power, Vertex-decomposability, Very well-covered graph, Well-covered graph",

author = "K. Kimura and Pournaki, {M. R.} and {Seyed Fakhari}, {S. A.} and N. Terai and S. Yassemi",

note = "Funding Information: The research of K. Kimura was in part supported by a grant from The Japan Society for the Promotion of Science (JSPS Grant-in-Aid for Young Scientists (B)—Ref. 15K17507). The research of M. R. Pournaki was in part supported by a grant from The World Academy of Sciences (TWAS–UNESCO Associateship—Ref. 3240295905). The research of N. Terai was in part supported by a grant from The Japan Society for the Promotion of Science (JSPS Grant-in-Aid for Scientific Research (C)—Ref. 18K03244) Publisher Copyright: {\textcopyright} 2022, The Author(s), under exclusive licence to Springer Nature Switzerland AG.",

year = "2022",

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doi = "10.1007/s40687-022-00326-2",

language = "English",

volume = "9",

journal = "Research in Mathematical Sciences",

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AU - Kimura, K.

AU - Pournaki, M. R.

AU - Seyed Fakhari, S. A.

AU - Terai, N.

AU - Yassemi, S.

N1 - Funding Information: The research of K. Kimura was in part supported by a grant from The Japan Society for the Promotion of Science (JSPS Grant-in-Aid for Young Scientists (B)—Ref. 15K17507). The research of M. R. Pournaki was in part supported by a grant from The World Academy of Sciences (TWAS–UNESCO Associateship—Ref. 3240295905). The research of N. Terai was in part supported by a grant from The Japan Society for the Promotion of Science (JSPS Grant-in-Aid for Scientific Research (C)—Ref. 18K03244) Publisher Copyright: © 2022, The Author(s), under exclusive licence to Springer Nature Switzerland AG.

PY - 2022/6

Y1 - 2022/6

N2 - A very well-covered graph is a well-covered graph without isolated vertices such that the height of its edge ideal is half of the number of vertices. In this survey article, we gather together most of the old and new results on the edge and cover ideals of these graphs.

AB - A very well-covered graph is a well-covered graph without isolated vertices such that the height of its edge ideal is half of the number of vertices. In this survey article, we gather together most of the old and new results on the edge and cover ideals of these graphs.

KW - Betti number

KW - CM property

KW - Cohen–Macaulay graph

KW - Cohen–Macaulay ring

KW - Compression

KW - Cover ideal

KW - Depth

KW - Edge ideal

KW - Edge-weighted ideal

KW - f-Vector

KW - Flag complex

KW - h-Vector

KW - Height

KW - Independence complex

KW - Linear resolution

KW - Local cohomology

KW - Minimal free resolution

KW - Projective dimension

KW - Regularity

KW - Shellability

KW - Simplicial complex

KW - Stanley–Reisner ideal

KW - Symbolic power

KW - Vertex-decomposability

KW - Very well-covered graph

KW - Well-covered graph

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A glimpse to most of the old and new results on very well-covered graphs from the viewpoint of commutative algebra

Abstract

Keywords

ASJC Scopus subject areas

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