scholarly journals A tight lower bound on the matching number of graphs via Laplacian eigenvalues

2022 ◽  
Vol 101 ◽  
pp. 103468
Author(s):  
Xiaofeng Gu ◽  
Muhuo Liu
Keyword(s):  
Lower Bound ◽  
Matching Number ◽  
Laplacian Eigenvalues

scholarly journals Lower bounds for fractional Laplacian eigenvalues

2014 ◽  
Vol 16 (06) ◽  
pp. 1450032
Author(s):  
Guoxin Wei ◽  
He-Jun Sun ◽  
Lingzhong Zeng
Keyword(s):  
Lower Bound ◽  
Euclidean Space ◽  
Bounded Domain ◽  
Lower Bounds ◽  
Fractional Laplacian ◽  
Dimensional Euclidean Space ◽  
Laplacian Eigenvalues ◽  
Eigenvalues Of The Laplacian ◽  
Eigenvalue Inequality

In this paper, we investigate eigenvalues of fractional Laplacian (–Δ)α/2|D, where α ∈ (0, 2], on a bounded domain in an n-dimensional Euclidean space and obtain a sharper lower bound for the sum of its eigenvalues, which improves some results due to Yildirim Yolcu and Yolcu in [Estimates for the sums of eigenvalues of the fractional Laplacian on a bounded domain, Commun. Contemp. Math. 15(3) (2013) 1250048]. In particular, for the case of Laplacian, we obtain a sharper eigenvalue inequality, which gives an improvement of the result due to Melas in [A lower bound for sums of eigenvalues of the Laplacian, Proc. Amer. Math. Soc. 131 (2003) 631–636].

scholarly journals Induced Matchings and the v-Number of Graded Ideals

Mathematics ◽  
2021 ◽  
Vol 9 (22) ◽  
pp. 2860
Author(s):  
Gonzalo Grisalde ◽  
Enrique Reyes ◽  
Rafael H. Villarreal
Keyword(s):  
Lower Bound ◽  
Upper Bound ◽  
Matching Number ◽  
Edge Ideal ◽  
Induced Matching ◽  
Simplicial Partition ◽  
The Family ◽  
Induced Matchings ◽  
Edge Ring ◽  
Insight Into

We give a formula for the v-number of a graded ideal that can be used to compute this number. Then, we show that for the edge ideal I(G) of a graph G, the induced matching number of G is an upper bound for the v-number of I(G) when G is very well-covered, or G has a simplicial partition, or G is well-covered connected and contains neither four, nor five cycles. In all these cases, the v-number of I(G) is a lower bound for the regularity of the edge ring of G. We classify when the induced matching number of G is an upper bound for the v-number of I(G) when G is a cycle and classify when all vertices of a graph are shedding vertices to gain insight into the family of W2-graphs.

Extremal graph on normalized Laplacian spectral radius and energy

2015 ◽  
Vol 29 ◽  
pp. 237-253 ◽  
Author(s):  
Kinkar Das ◽  
SHAOWEI SUN
Keyword(s):  
Lower Bound ◽  
Spectral Radius ◽  
Complete Graph ◽  
Connected Graph ◽  
Simple Graph ◽  
Extremal Graphs ◽  
Laplacian Spectral Radius ◽  
Laplacian Eigenvalues ◽  
Laplacian Energy ◽  
Isolated Vertex

Let $G=(V,\,E)$ be a simple graph of order $n$ and the normalized Laplacian eigenvalues $\rho_1\geq \rho_2\geq \cdots\geq\rho_{n-1}\geq \rho_n=0$. The normalized Laplacian energy (or Randi\'c energy) of $G$ without any isolated vertex is defined as $$RE(G)=\sum_{i=1}^{n}|\rho_i-1|.$$ In this paper, a lower bound on $\rho_1$ of connected graph $G$ ($G$ is not isomorphic to complete graph) is given and the extremal graphs (that is, the second minimal normalized Laplacian spectral radius of connected graphs) are characterized. Moreover, Nordhaus-Gaddum type results for $\rho_1$ are obtained. Recently, Gutman et al.~gave a conjecture on Randi\'c energy of connected graph [I. Gutman, B. Furtula, \c{S}. B. Bozkurt, On Randi\'c energy, Linear Algebra Appl. 442 (2014) 50--57]. Here this conjecture for starlike trees is proven.

scholarly journals More on the normalized Laplacian Estrada index

2014 ◽  
Vol 8 (2) ◽  
pp. 346-357 ◽  
Author(s):  
Yilun Shang
Keyword(s):  
Lower Bound ◽  
Bipartite Graphs ◽  
Simple Graph ◽  
Laplacian Eigenvalues ◽  
Estrada Index ◽  
General Graphs

Let G be a simple graph of order N. The normalized Laplacian Estrada index of G is defined as NEE(G)=?Ni=1 e?i?1, where ?1, ?2,... , ?N are the normalized Laplacian eigenvalues of G. In this paper, we give a tight lower bound for NEE of general graphs. We also calculate NEE for a class of treelike fractals, which contains T fractal and Peano basin fractal as its limiting cases. It is shown that NEE scales linearly with the order of the fractal, in line with a best possible lower bound for connected bipartite graphs.

scholarly journals A lower bound for the Laplacian eigenvalues of a graph—Proof of a conjecture by Guo

2008 ◽  
Vol 429 (8-9) ◽  
pp. 2131-2135 ◽  
Author(s):  
Andries E. Brouwer ◽  
Willem H. Haemers
Keyword(s):  
Lower Bound ◽  
Laplacian Eigenvalues

scholarly journals Matching Numbers and the Regularity of the Rees Algebra of an Edge Ideal

2020 ◽  
Vol 24 (3) ◽  
pp. 577-586
Author(s):  
Jürgen Herzog ◽  
Takayuki Hibi
Keyword(s):  
Lower Bound ◽  
Simple Graph ◽  
Matching Number ◽  
Edge Ideal ◽  
Rees Algebra ◽  
Induced Matching

Abstract The regularity $${\text {reg}}R(I(G))$$ reg R ( I ( G ) ) of the Rees ring R(I(G)) of the edge ideal I(G) of a finite simple graph G is studied. It is shown that, if R(I(G)) is normal, one has $${\text {mat}}(G) \le {\text {reg}}R(I(G)) \le {\text {mat}}(G) + 1$$ mat ( G ) ≤ reg R ( I ( G ) ) ≤ mat ( G ) + 1 , where $${\text {mat}}(G)$$ mat ( G ) is the matching number of G. In general, the induced matching number is a lower bound for the regularity, which can be shown by applying the squarefree divisor complex.

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