scholarly journals A Lower Bound for the Spectral Radius of Graphs with Fixed Diameter

2008 ◽  
Author(s):  
Sebastian Cioaba ◽  
Edwin van Dam ◽  
Jack Koolen ◽  
Jae-Ho Lee
Keyword(s):  
Lower Bound ◽  
Spectral Radius

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 Cauchy's interlace theorem and lower bounds for the spectral radius

2000 ◽  
Vol 23 (8) ◽  
pp. 563-566 ◽  
Author(s):  
A. McD. Mercer ◽  
Peter R. Mercer
Keyword(s):  
Lower Bound ◽  
Spectral Radius ◽  
Lower Bounds ◽  
Simple Proof ◽  
Symmetric Matrix ◽  
Real Symmetric Matrix

We present a short and simple proof of the well-known Cauchy interlace theorem. We use the theorem to improve some lower bound estimates for the spectral radius of a real symmetric matrix.

scholarly journals A lower bound for the spectral radius of graphs with fixed diameter

2010 ◽  
Vol 31 (6) ◽  
pp. 1560-1566 ◽  
Author(s):  
Sebastian M. Cioabă ◽  
Edwin R. van Dam ◽  
Jack H. Koolen ◽  
Jae-Ho Lee
Keyword(s):  
Lower Bound ◽  
Spectral Radius

scholarly journals Kesten’s theorem for uniformly recurrent subgroups

2019 ◽  
Vol 40 (10) ◽  
pp. 2778-2787
Author(s):  
MIKOLAJ FRACZYK
Keyword(s):  
Lower Bound ◽  
Cayley Graph ◽  
Spectral Radius ◽  
Short Proof ◽  
Ramanujan Graphs ◽  
Ramanujan Graph ◽  
Random Trajectory ◽  
The Difference

We prove a lower bound on the difference between the spectral radius of the Cayley graph of a group $G$ and the spectral radius of the Schreier graph $H\backslash G$ for any subgroup $H$. As an application, we extend Kesten’s theorem on spectral radii to uniformly recurrent subgroups and give a short proof that the result of Lyons and Peres on cycle density in Ramanujan graphs [Lyons and Peres. Cycle density in infinite Ramanujan graphs. Ann. Probab.43(6) (2015), 3337–3358, Theorem 1.2] holds on average. More precisely, we show that if ${\mathcal{G}}$ is an infinite deterministic Ramanujan graph then the time spent in short cycles by a random trajectory of length $n$ is $o(n)$.

scholarly journals A NOTE ON ENTROPY OF AUTO-EQUIVALENCES: LOWER BOUND AND THE CASE OF ORBIFOLD PROJECTIVE LINES

2018 ◽  
Vol 238 ◽  
pp. 86-103 ◽  
Author(s):  
KOHEI KIKUTA ◽  
YUUKI SHIRAISHI ◽  
ATSUSHI TAKAHASHI
Keyword(s):  
Lower Bound ◽  
Topological Entropy ◽  
Spectral Radius ◽  
Fundamental Theorem ◽  
Grothendieck Group ◽  
General Setting ◽  
Projective Lines

Entropy of categorical dynamics is defined by Dimitrov–Haiden–Katzarkov–Kontsevich. Motivated by the fundamental theorem of the topological entropy due to Gromov–Yomdin, it is natural to ask an equality between the entropy and the spectral radius of induced morphisms on the numerical Grothendieck group. In this paper, we add two results on this equality: the lower bound in a general setting and the equality for orbifold projective lines.

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).

Sign in / Sign up

Export Citation Format

Share Document