scholarly journals A lower bound on the spectral radius of the universal cover of a graph

2005 ◽  
Vol 93 (1) ◽  
pp. 33-43 ◽  
Author(s):  
Shlomo Hoory
Keyword(s):  
Lower Bound ◽  
Spectral Radius ◽  
Universal Cover

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 Slow volume growth for Reeb flows on spherizations and contact Bott–Samelson theorems

2015 ◽  
Vol 07 (03) ◽  
pp. 407-451 ◽  
Author(s):  
Urs Frauenfelder ◽  
Clémence Labrousse ◽  
Felix Schlenk
Keyword(s):  
Lower Bound ◽  
Polynomial Growth ◽  
Polynomial Complexity ◽  
Cohomology Ring ◽  
Volume Growth ◽  
Universal Cover ◽  
Geodesic Flows ◽  
Dynamical Complexity ◽  
Reeb Flows ◽  
Rank One

We give a uniform lower bound for the polynomial complexity of Reeb flows on the spherization (S*M, ξ) over a closed manifold. Our measure for the dynamical complexity of Reeb flows is slow volume growth, a polynomial version of topological entropy, and our lower bound is in terms of the polynomial growth of the homology of the based loop space of M. As an application, we extend the Bott–Samelson theorem from geodesic flows to Reeb flows: If (S*M, ξ) admits a periodic Reeb flow, or, more generally, if there exists a positive Legendrian loop of a fiber [Formula: see text], then M is a circle or the fundamental group of M is finite and the integral cohomology ring of the universal cover of M agrees with that of a compact rank one symmetric space.

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.

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