Results 1  10
of
13
Vibration Modes of 3ngaskets and other fractals
 J. PHYS. A: MATH THEOR
"... We study eigenvalues and eigenfunctions (vibration modes) on the class of selfsimilar symmetric finitely ramified fractals which includes 3ngaskets. We consider such examples as the Sierpinski gasket, a nonp.c.f. analog of the Sierpinski gasket, the level3 Sierpinski gasket, a fractal 3tree, t ..."
Abstract

Cited by 24 (12 self)
 Add to MetaCart
We study eigenvalues and eigenfunctions (vibration modes) on the class of selfsimilar symmetric finitely ramified fractals which includes 3ngaskets. We consider such examples as the Sierpinski gasket, a nonp.c.f. analog of the Sierpinski gasket, the level3 Sierpinski gasket, a fractal 3tree, the hexagasket, and one dimensional fractals. We develop a theoretical matrix analysis, including analysis of singularities, which allows us to compute eigenvalues, eigenfunctions and their multiplicities exactly. We support our theoretical analysis by symbolic and numerical computations.
ASYMPTOTICS OF THE TRANSITION PROBABILITIES OF THE SIMPLE RANDOM WALK ON SELFSIMILAR GRAPHS
, 2002
"... It is shown explicitly how selfsimilar graphs can be obtained as ‘blowup ’ constructions of finite cell graphs Ĉ. This yields a larger family of graphs than the graphs obtained by discretising continuous selfsimilar fractals. For a class of symmetrically selfsimilar graphs we study the simple ra ..."
Abstract

Cited by 19 (3 self)
 Add to MetaCart
It is shown explicitly how selfsimilar graphs can be obtained as ‘blowup ’ constructions of finite cell graphs Ĉ. This yields a larger family of graphs than the graphs obtained by discretising continuous selfsimilar fractals. For a class of symmetrically selfsimilar graphs we study the simple random walk on a cell graph Ĉ, starting in a vertex v of the boundary of Ĉ. It is proved that the expected number of returns to v before hitting another vertex in the boundary coincides with the resistance scaling factor. Using techniques from complex rational iteration and singularity analysis for Green functions we compute the asymptotic behaviour of the nstep transition probabilities of the simple random walk on the whole graph. The results of Grabner and Woess for the Sierpiński graph are generalised to the class of symmetrically selfsimilar graphs and at the same time the error term of the asymptotic expression is improved. Finally we present a criterion for the occurrence of oscillating phenomena of the nstep transition probabilities.
L²spectral invariants and convergent sequences of finite graphs
 JOURNAL OF FUNCTIONAL ANALYSIS
, 2008
"... Using the spectral theory of weakly convergent sequences of finite graphs, we prove the uniform existence of the integrated density of states for a large class of infinite graphs. ..."
Abstract

Cited by 13 (2 self)
 Add to MetaCart
(Show Context)
Using the spectral theory of weakly convergent sequences of finite graphs, we prove the uniform existence of the integrated density of states for a large class of infinite graphs.
A trace on fractal graphs AND THE IHARA ZETA FUNCTION
, 2008
"... Starting with Ihara’s work in 1968, there has been a growing interest in the study of zeta functions of finite graphs, by Sunada, Hashimoto, Bass, Stark and Terras, Mizuno and Sato, to name just a few authors. Then, Clair and MokhtariSharghi have studied zeta functions for infinite graphs acted u ..."
Abstract

Cited by 13 (5 self)
 Add to MetaCart
(Show Context)
Starting with Ihara’s work in 1968, there has been a growing interest in the study of zeta functions of finite graphs, by Sunada, Hashimoto, Bass, Stark and Terras, Mizuno and Sato, to name just a few authors. Then, Clair and MokhtariSharghi have studied zeta functions for infinite graphs acted upon by a discrete group of automorphisms. The main formula in all these treatments establishes a connection between the zeta function, originally defined as an infinite product, and the Laplacian of the graph. In this article, we consider a different class of infinite graphs. They are fractal graphs, i.e. they enjoy a selfsimilarity property. We define a zeta function for these graphs and, using the machinery of operator algebras, we prove a determinant formula, which relates the zeta function with the Laplacian of the graph. We also prove functional equations, and a formula which allows approximation of the zeta function by the zeta functions of finite subgraphs. The Ihara zeta function, originally associated to certain groups and then combinatorially
ENUMERATION OF MATCHINGS IN FAMILIES OF SELFSIMILAR GRAPHS
"... Abstract. The number of matchings of a graph G is an important graph parameter in various contexts, notably in statistical physics (dimermonomer model). Following recent research on graph parameters of this type in connection with selfsimilar, fractallike graphs, we study the asymptotic behavior ..."
Abstract

Cited by 5 (0 self)
 Add to MetaCart
(Show Context)
Abstract. The number of matchings of a graph G is an important graph parameter in various contexts, notably in statistical physics (dimermonomer model). Following recent research on graph parameters of this type in connection with selfsimilar, fractallike graphs, we study the asymptotic behavior of the number of matchings in families of selfsimilar graphs that are constructed by a very general replacement procedure. Under certain conditions on the geometry of the graphs, we are able to prove that the number of matchings generally follows a doubly exponential growth. The proof depends on an independence theorem for the number of matchings that has been used earlier to treat the special case of Sierpiński graphs. We provide a variety of examples and also discuss the situation when our conditions are not satisfied. 1.
The number of spanning trees in selfsimilar graphs
, 2008
"... Abstract. The number of spanning trees of a graph, also known as the complexity, is investigated for graphs which are constructed by a replacement procedure yielding a selfsimilar structure. It is shown that exact formulæ for the number of spanning trees can be given for sequences of selfsimilar g ..."
Abstract

Cited by 5 (3 self)
 Add to MetaCart
(Show Context)
Abstract. The number of spanning trees of a graph, also known as the complexity, is investigated for graphs which are constructed by a replacement procedure yielding a selfsimilar structure. It is shown that exact formulæ for the number of spanning trees can be given for sequences of selfsimilar graphs under certain symmetry conditions. These formulæ exhibit interesting connections to the theory of electrical networks. Examples include the wellknown Sierpiński graphs and their higherdimensional analoga. Several remarkable auxiliary results are provided on the way—for instance, a property of the number of rooted spanning forests is proven for graphs with a high degree of symmetry.
Quantifying the degree of selfnestedness of trees. Application to the structural analysis of plants
, 2009
"... apport de recherche ISSN 02496399 ISRN INRIA/RR6800FR+ENGinria00353645, version 1 16 Jan 2009Quantifying the degree of selfnestedness of trees. Application to the structural analysis of plants ..."
Abstract

Cited by 5 (3 self)
 Add to MetaCart
(Show Context)
apport de recherche ISSN 02496399 ISRN INRIA/RR6800FR+ENGinria00353645, version 1 16 Jan 2009Quantifying the degree of selfnestedness of trees. Application to the structural analysis of plants
Consensus and Coherence in Fractal Networks
, 2014
"... We consider first and second order consensus algorithms in networks with stochastic disturbances. We quantify the deviation from consensus using the notion of network coherence, which can be expressed as an H2 norm of the stochastic system. We use the setting of fractal networks to investigate the ..."
Abstract
 Add to MetaCart
We consider first and second order consensus algorithms in networks with stochastic disturbances. We quantify the deviation from consensus using the notion of network coherence, which can be expressed as an H2 norm of the stochastic system. We use the setting of fractal networks to investigate the question of whether a purely topological measure, such as the fractal dimension, can capture the asymptotics of coherence in the large system size limit. Our analysis for firstorder systems is facilitated by connections between firstorder stochastic consensus and the global mean first passage time of random walks. We then show how to apply similar techniques to analyze secondorder stochastic consensus systems. Our analysis reveals that two networks with the same fractal dimension can exhibit different asymptotic scalings for network coherence. Thus, this topological characterization of the network does not uniquely determine coherence behavior. The question of whether the performance of stochastic consensus algorithms in large networks can be captured by purely topological measures, such as the spatial dimension, remains open.
GreenFunctions on SelfSimilar Graphs and BOUNDS FOR THE SPECTRUM OF THE LAPLACIAN
, 2001
"... Combining the study of the simple random walk on graphs, generating functions (especially Green functions), complex dynamics and general complex analysis we introduce a new method of spectral analysis on selfsimilar graphs. We give an axiomatic definition of selfsimilar graphs which correspond to ..."
Abstract
 Add to MetaCart
Combining the study of the simple random walk on graphs, generating functions (especially Green functions), complex dynamics and general complex analysis we introduce a new method of spectral analysis on selfsimilar graphs. We give an axiomatic definition of selfsimilar graphs which correspond to general nested but not necessarily finitely ramified fractals. For this class of graphs a graph theoretic analogue to the Banach fixed point theorem is proved. Functional equations and a decomposition algorithm for the Green functions of selfsimilar graphs with some more symmetric structure are obtained. Their analytic continuations are given by rapidly converging expressions. We study the dynamics of a certain complex rational Green function d on finite directed subgraphs. If the Julia set J of d is a Cantor set, then the reciprocal spectrum spec −1 P = {1/z  z ∈ spec P} of the Markov transition operator P can be identified with the set of singularities of any Green function of the whole graph. Finally we get explicit upper and lower bounds for the reciprocal spectrum, where D is a countable set of the dbackwards iterates of a certain finite set of real numbers.
GREEN FUNCTIONS ON SELFSIMILAR GRAPHS AND BOUNDS FOR THE SPECTRUM OF THE LAPLACIAN
, 2002
"... Abstract. Combining the study of the simple random walk on graphs, generating functions (especially Green functions), complex dynamics and general complex analysis we introduce a new method of spectral analysis on selfsimilar graphs. We give an axiomatic definition of selfsimilar graphs which corr ..."
Abstract
 Add to MetaCart
(Show Context)
Abstract. Combining the study of the simple random walk on graphs, generating functions (especially Green functions), complex dynamics and general complex analysis we introduce a new method of spectral analysis on selfsimilar graphs. We give an axiomatic definition of selfsimilar graphs which correspond to general nested but not necessarily finitely ramified fractals. For this class of graphs a graph theoretic analogue to the Banach fixed point theorem is proved. Functional equations and a decomposition algorithm for the Green functions of selfsimilar graphs with some more symmetric structure are obtained. Their analytic continuations are given by rapidly converging expressions. We study the dynamics of a certain complex rational Green function d on finite directed subgraphs. If the Julia set J of d is a Cantor set, then the reciprocal spectrum spec −1 P = {1/z  z ∈ spec P} of the Markov transition operator P can be identified with the set of singularities of any Green function of the whole graph. Finally we get explicit upper and lower bounds for the reciprocal spectrum, where D is a countable set of the dbackwards iterates of a certain finite set of real numbers. 1.