A tight upper bound on the spectral radius of bottleneck matrices for graphs

2018 ◽  
Vol 551 ◽  
pp. 1-17
Author(s):  
Jason J. Molitierno
Keyword(s):  
Spectral Radius ◽  
Upper Bound

scholarly journals New Bounds for the α-Indices of Graphs

Mathematics ◽  
2020 ◽  
Vol 8 (10) ◽  
pp. 1668
Author(s):  
Eber Lenes ◽  
Exequiel Mallea-Zepeda ◽  
Jonnathan Rodríguez
Keyword(s):  
Lower Bound ◽  
Spectral Radius ◽  
Upper Bound ◽  
Adjacency Matrix ◽  
Independence Number ◽  
Minimal Degree ◽  
Diagonal Matrix ◽  
The Matrix

Let G be a graph, for any real 0≤α≤1, Nikiforov defines the matrix Aα(G) as Aα(G)=αD(G)+(1−α)A(G), where A(G) and D(G) are the adjacency matrix and diagonal matrix of degrees of the vertices of G. This paper presents some extremal results about the spectral radius ρα(G) of the matrix Aα(G). In particular, we give a lower bound on the spectral radius ρα(G) in terms of order and independence number. In addition, we obtain an upper bound for the spectral radius ρα(G) in terms of order and minimal degree. Furthermore, for n>l>0 and 1≤p≤⌊n−l2⌋, let Gp≅Kl∨(Kp∪Kn−p−l) be the graph obtained from the graphs Kl and Kp∪Kn−p−l and edges connecting each vertex of Kl with every vertex of Kp∪Kn−p−l. We prove that ρα(Gp+1)<ρα(Gp) for 1≤p≤⌊n−l2⌋−1.

scholarly journals Spectral Extrema for Graphs: The Zarankiewicz Problem

10.37236/212 ◽  
2009 ◽  
Vol 16 (1) ◽  
Author(s):  
László Babai ◽  
Barry Guiduli
Keyword(s):  
Bipartite Graph ◽  
Spectral Radius ◽  
Upper Bound ◽  
Adjacency Matrix ◽  
Complete Bipartite Graph ◽  
Average Degree ◽  
Largest Eigenvalue ◽  
Zarankiewicz Problem ◽  
The Largest Eigenvalue

Let $G$ be a graph on $n$ vertices with spectral radius $\lambda$ (this is the largest eigenvalue of the adjacency matrix of $G$). We show that if $G$ does not contain the complete bipartite graph $K_{t ,s}$ as a subgraph, where $2\le t \le s$, then $$\lambda \le \Big((s-1)^{1/t }+o(1)\Big)n^{1-1/t }$$ for fixed $t$ and $s$ while $n\to\infty$. Asymptotically, this bound matches the Kővári-Turán-Sós upper bound on the average degree of $G$ (the Zarankiewicz problem).

TWO SHARP UPPER BOUNDS FOR THE SIGNLESS LAPLACIAN SPECTRAL RADIUS OF GRAPHS

2011 ◽  
Vol 03 (02) ◽  
pp. 185-191 ◽  
Author(s):  
YA-HONG CHEN ◽  
RONG-YING PAN ◽  
XIAO-DONG ZHANG
Keyword(s):  
Spectral Radius ◽  
Upper Bound ◽  
Adjacency Matrix ◽  
Laplacian Matrix ◽  
Upper Bounds ◽  
Extremal Graphs ◽  
Laplacian Spectral Radius ◽  
Signless Laplacian Matrix ◽  
Signless Laplacian Spectral Radius ◽  
Signless Laplacian

The signless Laplacian matrix of a graph is the sum of its degree diagonal and adjacency matrices. In this paper, we present a sharp upper bound for the spectral radius of the adjacency matrix of a graph. Then this result and other known results are used to obtain two new sharp upper bounds for the signless Laplacian spectral radius. Moreover, the extremal graphs which attain an upper bound are characterized.

scholarly journals Properties and Applications of the Eigenvector Corresponding to the Laplacian Spectral Radius of a Graph

2013 ◽  
Vol 2013 ◽  
pp. 1-9 ◽  
Author(s):  
Haizhou Song ◽  
Qiufen Wang
Keyword(s):  
Spectral Radius ◽  
Upper Bound ◽  
Laplacian Spectrum ◽  
Laplacian Spectral Radius

We study the properties of the eigenvector corresponding to the Laplacian spectral radius of a graph and show some applications. We obtain some results on the Laplacian spectral radius of a graph by grafting and adding edges. We also determine the structure of the maximal Laplacian spectrum tree among trees withnvertices andkpendant vertices (n,kfixed), and the upper bound of the Laplacian spectral radius of some trees.

scholarly journals Sharp Bounds on the Spectral Radius of Nonnegative Matrices and Comparison to the Frobenius’ Bounds

2020 ◽  
Vol 14 ◽  
Keyword(s):  
Lower Bound ◽  
Spectral Radius ◽  
Discrete Time ◽  
Upper Bound ◽  
Nonnegative Matrix ◽  
Nonnegative Matrices ◽  
Sharp Bounds ◽  
Similarity Transformations ◽  
Linear Invariant ◽  
The Stability

In this paper, a new upper bound and a new lower bound for the spectral radius of a nοnnegative matrix are proved by using similarity transformations. These bounds depend only on the elements of the nonnegative matrix and its row sums and are compared to the well-established upper and lower Frobenius’ bounds. The proposed bounds are always sharper or equal to the Frobenius’ bounds. The conditions under which the new bounds are sharper than the Frobenius' ones are determined. Illustrative examples are also provided in order to highlight the sharpness of the proposed bounds in comparison with the Frobenius’ bounds. An application to linear invariant discrete-time nonnegative systems is given and the stability of the systems is investigated. The proposed bounds are computed with complexity O(n2).

scholarly journals A refined bound for the Z1-spectral radius of tensors

Filomat ◽  
2020 ◽  
Vol 34 (7) ◽  
pp. 2123-2129
Author(s):  
Yajun Liu ◽  
Chaoqian Li ◽  
Yaotang Li
Keyword(s):  
Uniform Distribution ◽  
Poisson Distribution ◽  
Spectral Radius ◽  
Gaussian Distribution ◽  
Upper Bound ◽  
Binomial Distribution ◽  
Numerical Experiments ◽  
Applied Mathematics

A refined upper bound for the Z1-spectral radius of tensors is given, which needs less computations than that presented by Wang et al. in [Applied Mathematics and Computation, 329 (2018) 266-277]. Numerical experiments involving Uniform distribution, Gaussian distribution, Poisson distribution and Binomial distribution are given to show the effectiveness of the proposed bound

scholarly journals An upper bound for the Z-spectral radius of adjacency tensors

2018 ◽  
Vol 2018 (1) ◽  
Author(s):  
Zhi-Yong Wu ◽  
Jun He ◽  
Yan-Min Liu ◽  
Jun-Kang Tian
Keyword(s):  
Spectral Radius ◽  
Upper Bound
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