113 research outputs found
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
A symmetric entropy bound on the non-reconstruction regime of Markov chains on Galton-Watson trees
We give a criterion of the form Q(d)c(M)<1 for the non-reconstructability of
tree-indexed q-state Markov chains obtained by broadcasting a signal from the
root with a given transition matrix M. Here c(M) is an explicit function, which
is convex over the set of M's with a given invariant distribution, that is
defined in terms of a (q-1)-dimensional variational problem over symmetric
entropies. Further Q(d) is the expected number of offspring on the
Galton-Watson tree. This result is equivalent to proving the extremality of the
free boundary condition-Gibbs measure within the corresponding Gibbs-simplex.
Our theorem holds for possibly non-reversible M and its proof is based on a
general Recursion Formula for expectations of a symmetrized relative entropy
function, which invites their use as a Lyapunov function.
In the case of the Potts model, the present theorem reproduces earlier
results of the authors, with a simplified proof, in the case of the symmetric
Ising model (where the argument becomes similar to the approach of Pemantle and
Peres) the method produces the correct reconstruction threshold), in the case
of the (strongly) asymmetric Ising model where the Kesten-Stigum bound is known
to be not sharp the method provides improved numerical bounds.Comment: 10 page
Generalized Hot Attractors
Non-extremal black holes are endowed with geometric invariants related to
their horizon areas. We extend earlier work on hot attractor black holes to
higher dimensions and add a scalar potential. In addition to the event and
Cauchy horizons, when we complexify the radial coordinate, non-extremal black
holes will generically have other horizons as well. We prove that the product
of all of the horizon areas is independent of variations of the asymptotic
moduli further generalizing the attractor mechanism for extremal black holes.
In the presence of a scalar potential, as typically appears in gauged
supergravity, we find that the product of horizon areas is not necessarily the
geometric mean of the extremal area, however. We outline the derivation of
horizon invariants for stationary backgrounds.Comment: 39 pages, 3 figures, v2 references and clarifications adde
Relating vertex and global graph entropy in randomly generated graphs
Combinatoric measures of entropy capture the complexity of a graph but rely upon the calculation of its independent sets, or collections of non-adjacent vertices. This decomposition of the vertex set is a known NP-Complete problem and for most real world graphs is an inaccessible calculation. Recent work by Dehmer et al. and Tee et al. identified a number of vertex level measures that do not suffer from this pathological computational complexity, but that can be shown to be effective at quantifying graph complexity. In this paper, we consider whether these local measures are fundamentally equivalent to global entropy measures. Specifically, we investigate the existence of a correlation between vertex level and global measures of entropy for a narrow subset of random graphs. We use the greedy algorithm approximation for calculating the chromatic information and therefore Körner entropy. We are able to demonstrate strong correlation for this subset of graphs and outline how this may arise theoretically
- …