On Betti numbers of edge ideals of crown graphs
type: Item
instance of: scholarly article
title: On Betti numbers of edge ideals of crown graphs
zbMATH Open document ID: 1408.13038
author: Shahnawaz Ahmad Rather
author: Pavinder Singh
published in: Beiträge zur Algebra und Geometrie
publication date: 2019-02-26T00:00:00Z
review text: Let $G=(V, E)$ be a finite simple (no loops or multiple edges) undirected graph with vertex set $V(G)=\{x_1, x_2,\dots, x_n\}$ and edge set $E(G)$. Let $I(G)=(x_ix_j\mid \{x_i,x_j\}\in E(G))$ be an ideal in the polynomial ring $R=\mathbb{K}[x_1,\dots,x_n]$, where the vertex $x_i$ is identified with the variable $x_i$. The ideal $I(G)$ is called the edge ideal of $G$. \par The authors study edge ideals mainly to investigate relations between algebraic properties of edge ideals and combinatorial properties of graphs. They mainly focus on describing invariants of $I(G)$ in terms of $G$. The organization of the paper is as follows: In Sect. 2 the authors recall some definitions and introduce the notion of the strongly disjoint set of bouquets. Moreover, in this section, they recall some well-known results by \textit{M. Hochster} [in: Proc. Second Conf., Univ. Oklahoma, Norman, Okla. 171--223 (1975; Zbl 0351.13009)] and \textit{M. Katzman} [J. Comb. Theory, Ser. A 113, No. 3, 435--454 (2006; Zbl 1102.13024)]. In Sect. 3 the authors introduce the crown graphs and prove some results. In particular, they prove Theorems 3.1 and 3.9. In Sect. 4 the authors discuss domination parameters of graphs and present a result on projective dimension of edge ideals of crown graphs.
reviewed by: Amir Mafi
Mathematics Subject Classification ID: 13D02
Mathematics Subject Classification ID: 13F55
Mathematics Subject Classification ID: 05C69
Mathematics Subject Classification ID: 05C99
zbMATH DE Number: 7029753
zbMATH Keywords: crown graphs
zbMATH Keywords: Betti numbers
zbMATH Keywords: bouquets
zbMATH Keywords: projective dimension
zbMATH Keywords: regularity
describes a project that uses: Macaulay2
MaRDI profile type: Publication
full work available at URL: s13366-018-0397-3
OpenAlex ID: W2801745334
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Wikidata QID: Q129971630
DOI: 10.1007/S13366-018-0397-3
description: scientific article; zbMATH DE number 7029753