scholarly journals Regularity of binomial edge ideals of chordal graphs

2020 ◽  
Author(s):  
Mohammad Rouzbahani Malayeri ◽  
Sara Saeedi Madani ◽  
Dariush Kiani
Keyword(s):  
Chordal Graphs ◽  
Edge Ideals

scholarly journals An Upper Bound for the Regularity of Symbolic Powers of Edge Ideals of Chordal Graphs

10.37236/8566 ◽  
2019 ◽  
Vol 26 (2) ◽  
Author(s):  
Seyed Amin Seyed Fakhari
Keyword(s):  
Upper Bound ◽  
Chordal Graph ◽  
Chordal Graphs ◽  
Symbolic Power ◽  
Matching Number ◽  
Edge Ideal ◽  
Edge Ideals ◽  
Symbolic Powers ◽  
Reg I

Assume that $G$ is a chordal graph with edge ideal $I(G)$ and ordered matching number $\nu_{o}(G)$. For every integer $s\geq 1$, we denote the $s$-th symbolic power of $I(G)$ by $I(G)^{(s)}$. It is shown that ${\rm reg}(I(G)^{(s)})\leq 2s+\nu_{o}(G)-1$. As a consequence, we determine the regularity of symbolic powers of edge ideals of chordal Cameron-Walker graphs.

scholarly journals Certain algebraic invariants of edge ideals of join of graphs

2020 ◽  
pp. 2150099
Author(s):  
Arvind Kumar ◽  
Rajiv Kumar ◽  
Rajib Sarkar
Keyword(s):  
Simple Graph ◽  
Chordal Graphs ◽  
Edge Ideal ◽  
Edge Ideals ◽  
Symbolic Powers ◽  
Multipartite Graphs ◽  
Wheel Graphs ◽  
Join Of Graphs ◽  
Depth Dimension ◽  
Complete Multipartite

Let [Formula: see text] be a simple graph and [Formula: see text] be its edge ideal. In this paper, we study the Castelnuovo–Mumford regularity of symbolic powers of edge ideals of join of graphs. As a consequence, we prove Minh’s conjecture for wheel graphs, complete multipartite graphs, and a subclass of co-chordal graphs. We obtain a class of graphs whose edge ideals have regularity three. By constructing graphs, we prove that the multiplicity of edge ideals of graphs is independent from the depth, dimension, regularity, and degree of [Formula: see text]-polynomial. Also, we demonstrate that the depth of edge ideals of graphs is independent from the regularity and degree of [Formula: see text]-polynomial by constructing graphs.

scholarly journals A proof for a conjecture on the regularity of binomial edge ideals

2021 ◽  
Vol 180 ◽  
pp. 105432
Author(s):  
Mohammad Rouzbahani Malayeri ◽  
Sara Saeedi Madani ◽  
Dariush Kiani
Keyword(s):  
Edge Ideals

Chordal graphs

2021 ◽  
pp. 130-151
Author(s):  
Martin Charles Golumbic
Keyword(s):  
Chordal Graphs

On 3-Degree 4-Chordal Graphs

2020 ◽  
Author(s):  
Devarshi Aggarwal ◽  
R.Mahendra Kumar ◽  
Shwet Prakash ◽  
N. Sadagopan
Keyword(s):  
Chordal Graphs

scholarly journals On vanishing patterns in j-strands of edge ideals

2017 ◽  
Vol 46 (2) ◽  
pp. 287-295 ◽  
Author(s):  
Abed Abedelfatah ◽  
Eran Nevo
Keyword(s):  
Edge Ideals

scholarly journals Stanley depth of edge ideals

2012 ◽  
Vol 49 (4) ◽  
pp. 501-508 ◽  
Author(s):  
Muhammad Ishaq ◽  
Muhammad Qureshi
Keyword(s):  
Complete Graph ◽  
Upper Bound ◽  
Edge Ideal ◽  
Stanley Depth ◽  
Edge Ideals

We give an upper bound for the Stanley depth of the edge ideal I of a k-partite complete graph and show that Stanley’s conjecture holds for I. Also we give an upper bound for the Stanley depth of the edge ideal of a s-uniform complete bipartite hypergraph.

scholarly journals Gorenstein binomial edge ideals

2021 ◽  
Author(s):  
René González‐Martínez
Keyword(s):  
Edge Ideals

k-Efficient domination: Algorithmic perspective

2022 ◽  
Author(s):  
Mohsen Alambardar Meybodi
Keyword(s):  
Decision Problem ◽  
Dominating Set ◽  
Polynomial Time Algorithm ◽  
Time Algorithm ◽  
Chordal Graphs ◽  
Sparse Graphs ◽  
Fpt Algorithm ◽  
Domination Problem ◽  
Efficient Dominating Set ◽  
Np Complete

A set [Formula: see text] of a graph [Formula: see text] is called an efficient dominating set of [Formula: see text] if every vertex [Formula: see text] has exactly one neighbor in [Formula: see text], in other words, the vertex set [Formula: see text] is partitioned to some circles with radius one such that the vertices in [Formula: see text] are the centers of partitions. A generalization of this concept, introduced by Chellali et al. [k-Efficient partitions of graphs, Commun. Comb. Optim. 4 (2019) 109–122], is called [Formula: see text]-efficient dominating set that briefly partitions the vertices of graph with different radiuses. It leads to a partition set [Formula: see text] such that each [Formula: see text] consists a center vertex [Formula: see text] and all the vertices in distance [Formula: see text], where [Formula: see text]. In other words, there exist the dominators with various dominating powers. The problem of finding minimum set [Formula: see text] is called the minimum [Formula: see text]-efficient domination problem. Given a positive integer [Formula: see text] and a graph [Formula: see text], the [Formula: see text]-efficient Domination Decision problem is to decide whether [Formula: see text] has a [Formula: see text]-efficient dominating set of cardinality at most [Formula: see text]. The [Formula: see text]-efficient Domination Decision problem is known to be NP-complete even for bipartite graphs [M. Chellali, T. W. Haynes and S. Hedetniemi, k-Efficient partitions of graphs, Commun. Comb. Optim. 4 (2019) 109–122]. Clearly, every graph has a [Formula: see text]-efficient dominating set but it is not correct for efficient dominating set. In this paper, we study the following: [Formula: see text]-efficient domination problem set is NP-complete even in chordal graphs. A polynomial-time algorithm for [Formula: see text]-efficient domination in trees. [Formula: see text]-efficient domination on sparse graphs from the parametrized complexity perspective. In particular, we show that it is [Formula: see text]-hard on d-degenerate graphs while the original dominating set has Fixed Parameter Tractable (FPT) algorithm on d-degenerate graphs. [Formula: see text]-efficient domination on nowhere-dense graphs is FPT.

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