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

A glimpse to most of the old and new results on very well-covered graphs from the viewpoint of... 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. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Research in the Mathematical Sciences Springer Journals

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

, Volume 9 (2) – Jun 1, 2022
18 pages

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Publisher
Springer Journals
Copyright © The Author(s), under exclusive licence to Springer Nature Switzerland AG 2022
eISSN
2197-9847
DOI
10.1007/s40687-022-00326-2
Publisher site
See Article on Publisher Site

### 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.

### Journal

Research in the Mathematical SciencesSpringer Journals

Published: Jun 1, 2022

Keywords: Betti number; CMt\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathrm{CM}_t$$\end{document} property; Cohen–Macaulay graph; Cohen–Macaulay ring; Compression; Cover ideal; Depth; Edge ideal; Edge-weighted ideal; Flag complex; f-Vector; Height; h-Vector; 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; 05C75; 05C90; 13D45; 13F55; 13H10; 55U10

### References

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