20,220 research outputs found
Edge-disjoint spanning trees and eigenvalues of regular graphs
Partially answering a question of Paul Seymour, we obtain a sufficient
eigenvalue condition for the existence of edge-disjoint spanning trees in a
regular graph, when . More precisely, we show that if the second
largest eigenvalue of a -regular graph is less than
, then contains at least edge-disjoint spanning
trees, when . We construct examples of graphs that show our
bounds are essentially best possible. We conjecture that the above statement is
true for any .Comment: 4 figure
The Singular Values of the Exponientiated Adjacency Matrixes of Broom-Tree Graphs
In this paper, we explore the singular values of adjacency matrices {An} for a particular family {Gn} of graphs, known as broom trees. The singular values of a matrix M are defined to be the square roots of the eigenvalues of the symmetrized matrix MTM. The matrices we are interested in are the symmetrized adjacency matrices AnTAn and the symmetrized exponentiated adjacency matrices BnTBn = (eAn − I)T(eAn − I) of the graphs Gn. The application of these matrices in the HITS algorithm for Internet searches suggests that we study whether the largest two eigenvalues of AnTAn (or those of BnTBn) can become close or in fact coincide. We have shown that for one family of broom-trees, the ratio of the two largest eigenvalues of BnTBn as the number n of nodes (more specifically, the length l of the graph) goes to infinity is bounded below one. This bound shows that for these graphs, the second largest eigenvalue remains bounded away from the largest eigenvalue. For a second family of broom trees it is not known whether the same is true. However, we have shown that for that family a certain later eigenvalue remains bounded away from the largest eigenvalue. Our last result is a generalization of this latter result
Maximal Entropy Random Walk: solvable cases of dynamics
We focus on the study of dynamics of two kinds of random walk: generic random
walk (GRW) and maximal entropy random walk (MERW) on two model networks: Cayley
trees and ladder graphs. The stationary probability distribution for MERW is
given by the squared components of the eigenvector associated with the largest
eigenvalue \lambda_0 of the adjacency matrix of a graph, while the dynamics of
the probability distribution approaching to the stationary state depends on the
second largest eigenvalue \lambda_1.
Firstly, we give analytic solutions for Cayley trees with arbitrary branching
number, root degree, and number of generations. We determine three regimes of a
tree structure that result in different statics and dynamics of MERW, which are
due to strongly, critically, and weakly branched roots. We show how the
relaxation times, generically shorter for MERW than for GRW, scale with the
graph size.
Secondly, we give numerical results for ladder graphs with symmetric defects.
MERW shows a clear exponential growth of the relaxation time with the size of
defective regions, which indicates trapping of a particle within highly
entropic intact region and its escaping that resembles quantum tunneling
through a potential barrier. GRW shows standard diffusive dependence
irrespective of the defects.Comment: 13 pages, 6 figures, 24th Marian Smoluchowski Symposium on
Statistical Physics (Zakopane, Poland, September 17-22, 2011
Abelian Spiders
If G is a finite graph, then the largest eigenvalue L of the adjacency matrix
of G is a totally real algebraic integer (L is the Perron-Frobenius eigenvalue
of G). We say that G is abelian if the field generated by L^2 is abelian. Given
a fixed graph G and a fixed set of vertices of G, we define a spider graph to
be a graph obtained by attaching to each of the chosen vertices of G some
2-valent trees of finite length. The main result is that only finitely many of
the corresponding spider graphs are both abelian and not Dynkin diagrams, and
that all such spiders can be effectively enumerated; this generalizes a
previous result of Calegari, Morrison, and Snyder. The main theorem has
applications to the classification of finite index subfactors. We also prove
that the set of Salem numbers of "abelian type" is discrete.Comment: This work represents, in part, the PhD thesis of the second autho
The second largest eigenvalue of a tree
AbstractDenote by λ2(T) the second largest eigenvalue of a tree T. An easy algorithm is given to decide whether λ2(T)⩽λ for a given number λ, and a structure theorem for trees withλ2(T)⩽λ is proved. Also, it is shown that a tree T with n vertices has λ2(T)⩽lsqb(n−3)2rsqb12; this bound is best possible for odd n
Quadratic embedding constants of graphs: Bounds and distance spectra
The quadratic embedding constant (QEC) of a finite, simple, connected graph
is the maximum of the quadratic form of the distance matrix of on the
subset of the unit sphere orthogonal to the all-ones vector. The study of these
QECs was motivated by the classical work of Schoenberg on quadratic embedding
of metric spaces [Ann. of Math., 1935] and [Trans. Amer. Math. Soc., 1938]. In
this article, we provide sharp upper and lower bounds for the QEC of trees. We
next explore the relation between distance spectra and quadratic embedding
constants of graphs - and show two further results: We show that the
quadratic embedding constant of a graph is zero if and only if its second
largest distance eigenvalue is zero. We identify a new subclass of
nonsingular graphs whose QEC is the second largest distance eigenvalue.
Finally, we show that the QEC of the cluster of an arbitrary graph with
either a complete or star graph can be computed in terms of the QEC of . As
an application of this result, we provide new families of examples of graphs of
QE class.Comment: 15 pages, 2 figure
On the spectral properties of Feigenbaum graphs
A Horizontal Visibility Graph (HVG) is a simple graph extracted from an
ordered sequence of real values, and this mapping has been used to provide a
combinatorial encryption of time series for the task of performing network
based time series analysis. While some properties of the spectrum of these
graphs --such as the largest eigenvalue of the adjacency matrix-- have been
routinely used as measures to characterise time series complexity, a theoretic
understanding of such properties is lacking. In this work we explore some
algebraic and spectral properties of these graphs associated to periodic and
chaotic time series. We focus on the family of Feigenbaum graphs, which are
HVGs constructed in correspondence with the trajectories of one-parameter
unimodal maps undergoing a period-doubling route to chaos (Feigenbaum
scenario). For the set of values of the map's parameter for which the
orbits are periodic with period , Feigenbaum graphs are fully
characterised by two integers (n,k) and admit an algebraic structure. We
explore the spectral properties of these graphs for finite n and k, and among
other interesting patterns we find a scaling relation for the maximal
eigenvalue and we prove some bounds explaining it. We also provide numerical
and rigorous results on a few other properties including the determinant or the
number of spanning trees. In a second step, we explore the set of Feigenbaum
graphs obtained for the range of values of the map's parameter for which
the system displays chaos. We show that in this case, Feigenbaum graphs form an
ensemble for each value of and the system is typically weakly
self-averaging. Unexpectedly, we find that while the largest eigenvalue can
distinguish chaos from an iid process, it is not a good measure to quantify the
chaoticity of the process, and that the eigenvalue density does a better job.Comment: 33 page
Graph homomorphisms between trees
In this paper we study several problems concerning the number of
homomorphisms of trees. We give an algorithm for the number of homomorphisms
from a tree to any graph by the Transfer-matrix method. By using this algorithm
and some transformations on trees, we study various extremal problems about the
number of homomorphisms of trees. These applications include a far reaching
generalization of Bollob\'as and Tyomkyn's result concerning the number of
walks in trees.
Some other highlights of the paper are the following. Denote by
the number of homomorphisms from a graph to a graph . For any tree
on vertices we give a general lower bound for by certain
entropies of Markov chains defined on the graph . As a particular case, we
show that for any graph ,
where is the
largest eigenvalue of the adjacency matrix of and is a
certain constant depending only on which we call the spectral entropy of
. In the particular case when is the path on vertices, we
prove that where
is any tree on vertices, and and denote the path and star on
vertices, respectively. We also show that if is any fixed tree and
for some tree on vertices, then
must be the tree obtained from a path by attaching a pendant
vertex to the second vertex of .
All the results together enable us to show that
|\End(P_m)|\leq|\End(T_m)|\leq|\End(S_m)|, where \End(T_m) is the set of
all endomorphisms of (homomorphisms from to itself).Comment: 47 pages, 15 figure
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