Peculiar spectral statistics of ensembles of trees and star-like graphs
PPeculiar spectral statistics of ensembles of trees and star-like graphs
V. Kovaleva , Yu. Maximov , , S. Nechaev , , and O. Valba , Center of Energy Systems, Skolkovo Institute of Science and Technology, Russia Center for Nonlinear Studies and Theoretical Division T-4,Los Alamos National Laboratory, Los Alamos, NM 87545, USA CNRS/Independent University of Moscow, Poncelet Laboratory, Moscow, Russia P.N. Lebedev Physical Institute RAS, Moscow, Russia Department of Applied Mathematics, National Research University Higher School of Economics N.N. Semenov Institute of Chemical Physics of the Russian Academy of Sciences
In this paper we investigate the eigenvalue statistics of exponentially weighted ensembles of fullbinary trees and p -branching star graphs. We show that spectral densities of corresponding adja-cency matrices demonstrate peculiar ultrametric structure inherent to sparse systems. In particular,the tails of the distribution for binary trees share the “Lifshitz singularity” emerging in the one-dimensional localization, while the spectral statistics of p -branching star-like graphs is less universal,being strongly dependent on p . The hierarchical structure of spectra of adjacency matrices is inter-preted as sets of resonance frequencies, that emerge in ensembles of fully branched tree-like systems,known as dendrimers. However, the relaxational spectrum is not determined by the cluster topology,but has rather the number-theoretic origin, reflecting the peculiarities of the rare-event statisticstypical for one-dimensional systems with a quenched structural disorder. The similarity of spectraldensities of an individual dendrimer and of ensemble of linear chains with exponential distribu-tion in lengths, demonstrates that dendrimers could be served as simple disorder-less toy models ofone-dimensional systems with quenched disorder. I. INTRODUCTION
One can gain the information about topological and statistical properties of polymers in solutions by measuringtheir relaxation spectra [1]. A rough model of an individual polymer molecule of any topology is a set of monomers(atoms) connected by elastic strings. If deformations of strings are small, the response of the molecule on externalexcitation is harmonic according to the Hooke’s law. The relaxation modes are basically determined by the so-calledLaplacian matrix (defined below) of the molecule.Consider the polymer network as a graph or a collection of graphs, see Fig. 1a. Enumerate the monomers ofthe N -atomic macromolecule by the index i = 1 . . . N . The adjacency matrix A = { a ij } describing the topology(connectivity) of a polymer molecule is symmetric ( a ij = a ji ). Its matrix elements a ij take binary values, 0 and 1,such that the diagonal elements vanish, i.e. a ii = 0, and for off-diagonal elements, i (cid:54) = j , one has a ij = 1, if themonomers i and j are connected, and a ij = 0 otherwise. The Laplacian matrix L = { b ij } is by definition as follows: b ij = − a ij for i (cid:54) = j , and b ii = (cid:80) Nj =1 a ij , as shown in Fig. 1b, i.e. L = dI − A , where d is the vector of vertex degreesof the graph and I is the identity matrix. The eigenvalues λ n ( n = 1 , ..., N ) of the symmetric matrix L are real. Forregular graphs (i.e for graphs with constant vertex degrees) the spectra of the adjacency and Laplacian matrices areuniquely connected to each other. adjacency matrix -- -- - ---- -- -Laplacian matrix Figure 1: (a) Sample of a topological graph (collection of subgraphs) and its adjacency matrix; (b) Elastic networkcorresponding to (a) and its Laplacian matrix. a r X i v : . [ c ond - m a t . s t a t - m ec h ] M a y In physical literature the spectrum of adjacency matrix is interpreted as the set of resonance frequencies, whilethe Laplacian spectrum defines the relaxation of the system. Thus, measuring the response of the diluted solutionof noninteracting polymer molecules on external excitations with continuously changing wavelength, we expect to seethe signature of eigenmodes in the spectral density as peaks at some specific frequencies. Some other applications ofadjacency and Laplacian matrices in graph theory and optimization are thoroughly described in [2] and [3].Previously, in [4] the eigenvalue density in ensembles of large sparse random adjacency matrices was investigated.It has been demonstrated that the fraction of linear subgraphs at the percolation threshold is about 95% of all finitesubgraphs, and the distribution of linear subgraphs (chains) is purely exponential. Analyzing in detail the spectraldensity of linear chain ensembles, the authors claim in [4] its ultrametric nature and showed that near the edge of thespectrum, λ max , the tails of the spectral density, ρ ( λ ) exhibit a so-called “Lifshitz singularity”, ρ ( λ ) ∼ e − c/ √ λ max − λ ,typical for the one-dimensional Anderson localization citelifshitz. In [4] the attention was paid to a connection ofthe spectral density to the modular geometry and, in particular, to the Dedekind η -function. In this work it wasconjectured that ultrametricity emerges in any rare-event statistics and is inherit to generic complex sparse systems.The rare-event statistics has many manifestations in physics [6] and biophysics [7]. For example, the contact mapsof individual DNA molecules in cell nuclei are sparse. Thus, experimenting with physical properties of highly dilutedsolutions of biologically active substances, one should pay attention to a very peculiar structure of background noiseoriginating from the rare-event statistics of dissolved clusters. In this case, the peculiar shape of noise spectrum canbe misinterpreted or at least can make the data incomprehensible [8–10]. In order to make conclusions about anybiological activity of regarded chemical substance, the signal from background noise should be clearly identified. Fromthis point of view, the work [11] seems very interesting, since it represents an exceptional example of careful attentionto ultrametric-like distributions in biological and clinical data.In the current work we investigate in detail the spectral statistics (eigenvalue density of the adjacency matrices) ofbranched polymer ensembles: (i) complete three-branching trees, and (ii) p -branching star-like graphs. These struc-tures represent special cases of branching polymers, known as dendrimers [12]. Recall that dendrimers are regularlybranching macromolecules with a tree-like topology. Computing the eigenvalues (resonances) of the adjacency matrix,and the degeneracy of these resonances in the polymer ensemble, we provide the information about the excitationstatistics, analyze the tails of corresponding distributions and discuss the origin of clearly seen ultrametricity [13, 14].We obtain the results for adjacency matrices (but not for the Laplacian ones) to make our conclusions comparablewith former research dealt with linear chain ensembles. Special attention we would like to pay to the works [15, 16],where the relaxational spectra of dendrimer molecules have been analytically investigated and physical properties ofcorresponding dendrimers have been studied. Compared to the works [15, 16], the goal of our paper is in some sense,contraversal: we would like to emphasize that dendrimers, as well as ensembles of linear polymers, and ensembles ofsparse randomly branching clusters, share the generic hierarchical spectral statistics. From this point of view, thespectral statistics seen for dendrimers is not uniquely determined by the topology of the graphs under consideration.To make the content of the paper as self-consistent as possible, we remind briefly the notion of ultrametricity.A metric space is a set of elements equipped by pairwise distances, d ( x, y ) between elements x and y . The metric d ( x, y ) meets three requirements: i) non-negativity, d ( x, y ) > x (cid:54) = y , and d ( x, y ) = 0 for x = y , ii) symmetry, d ( x, y ) = d ( y, x ), and iii) the triangle inequality, d ( x, z ) ≤ d ( x, y ) + d ( y, z ). The concept of ultrametricity is relatedto a special class of metrics, obeying the strong triangular inequality , d ( x, z ) ≤ max { d ( x, y ) , d ( y, z ) } .The description of ultrametric systems in physical terms deals with the concept of hierarchical organization ofenergy landscapes [13, 14]. A complex system is assumed to have a large number of metastable states correspondingto local minima in the potential energy landscape. With respect to the transition rates, the minima are suggested tobe clustered in hierarchically nested basins, i.e. larger basins consist of smaller basins, each of those consists of evensmaller ones, etc . The basins of local energy minima are separated by a hierarchically arranged set of barriers: largebasins are separated by high barriers, and smaller basins within each larger one are separated by lower barriers.Since the transitions between the basins are determined by the passages over the highest barriers separating them,the transitions between any two local minima obey the strong triangle inequality. The ultrametric organization ofspectral densities obtained in our work should be understood exactly in that sense, if we identify the degeneracyof the eigenvalue with the height of the barrier separating some points on the spectral axis. Ultrametric geometryfixes taxonomic (i.e. hierarchical) tree-like relationships between elements and, speaking figuratively, is closer toLobachevsky geometry, rather than to the Euclidean one.The selection of binary trees and stars for our study is not occasional. We are interested in the question whetherthe hierarchical structure of spectral density emerges in polymer ensembles beyond the linear statistics. Moreover,ensembles of full binary trees and stars allow for complete analytic treatment, which makes them the first candidatesbeyond the rare-event statistics emerging in linear ensembles. Varying the branching number of star-like graphs, wecan interpolate between linear chains and branching structures, attempting to understand the influence of non-lineartopology on spectral statistics.It is essential to emphasize that we investigate ensembles of non-interacting polymer molecules. This is ensuredby the low density of polymers in the solution [17]. Understanding how the inter-molecular interactions change thespectral density of the polymer solution is a challenging problem which yet is beyond the scopes of our investigations,however definitely will be tackled later. Additional remarks should be made regarding the polymer size distributionin ensembles. It is crucial that we study polydisperse ensembles (i.e. ensemble of polymers of various sizes). Here wesuppose the probability P ( N ) to find an N -atomic chain, to be exponential, P ( N ) ∼ e − µN ( µ > II. GRAPHS UNDER CONSIDERATION: FULL BINARY TREES AND STARS
In [4] the authors discussed some statistical properties of polydisperse ensembles of linear macromolecules andpaid attention to two specific properties: i) the singularity of the enveloping curve of spectral density at the edgeof the spectrum, known as the “Lifshitz tail”, and ii) the hierarchical organization of resonances in the bulk of thespectrum. It was shown in [4] that these properties are inherent to generic sparse ensembles and can be viewed asnumber-theoretic manifestations of the rare-event statistics. Whether they survive for ensembles of trees or stars isthe question analyzed in present work.Below we compute eigenvalues with corresponding multiplicities (degeneracies) of adjacency matrices of full bi-nary trees and star-like graphs (dendrimers), schematically depicted in Figs. 2a,b. Then we perform averaging overensembles of trees of particular topology and determine the spectral density. ... ... (a) (b) k=1k=2k 1 2 m-1 m
Figure 2: (a) Full binary tree; (b) a star-like graph with p branches of k nodes.Let T k be a full binary tree of level k shown in the Fig. 2a. The shortest path from the root of T k to any leaf is k −
1, and the tree has 2 k − T k be the adjacency matrix of T k . The spectrum of an individualtree T k has exactly N treek = 2 k − ρ k ( λ ) : R → R be the spectral density of T k , ρ k ( λ ) = M k ( λ ) N treek , (1)where λ is an eigenvalue, and M k ( λ ) is its multiplicity. We investigate the ensemble of such trees with the exponentialdistribution, P µ ( k ) = Ce − µk , ∞ (cid:88) k =1 P µ ( k ) = 1 , (2)where C = e µ −
1, and µ is a parameter. The spectral density, ρ ( λ ) of the ensemble is the quotient of multiplicityexpectation and tree size expectation ρ ( λ ) = ∞ (cid:80) k =1 P µ ( k ) M k ( λ ) ∞ (cid:80) k =1 P µ ( k ) N treek . (3)For a star graph depicted in the Fig. 2b we state the problem similarly. Let S k,p be a star graph constructed bygluing p linear graphs (chains) at one point, all extended up to the level k . The graph has p ( k −
1) + 1 vertices intotal, and its adjacency matrix is S k,p with the total number of eigenvalues N stark,p = p ( k −
1) + 1. Let g k,p ( λ ) : R → R be the spectral density of S k,p , g k,p ( λ ) = M k,p ( λ ) N stark,p , (4)where λ is an eigenvalue and M k,p ( λ ) is its multiplicity. The probability of a star to have parameters k and p is P ( k | p ) P ( p ). Specifically, we consider the ensemble of stars with fixed p , where k is distributed as in (2): P µ ( k | p ) = Ce − µk , ∞ (cid:88) k =1 P µ ( k | p ) = 1 , (cid:88) p ≥ P µ ( p ) = 1 . (5)Thus, the spectral density g p ( λ ) of the p -branch star ensemble is g p ( λ ) = ∞ (cid:80) k =1 P µ ( k | p )M k,p ( λ ) ∞ (cid:80) k =1 P µ ( k | p ) N stark,p . (6)We suppose that p ≥
3, since p = 1 , III. RESULTS
Below we provide numeric and analytic results for spectral densities of dendrimer ensembles. Numeric simulationsallow us to get a visual representation of the investigated objects in order to make plausible conjectures aboutindividual graph spectra as well as to understand generic features of the corresponding spectral densities of thesystems under consideration.The main steps of the numeric algorithm are as follows. We calculate eigenvalues of T k or S k,p with correspondingmultiplicities. The computational complexity of eigenvalue calculation is O ( n ), where n the number of vertices. Notethat accumulation of small computational errors can lead to inaccurate results. That is why we construct a histograminstead of treating each eigenvalue separately. Specifically, we divide the axis of eigenvalues into the intervals of length∆ and construct the piecewise constant function ˆ f k ( λ ) : R → R . On a particular segment the function ˆ f k ( λ ) is equalto the number of eigenvalues in this segment:ˆ f k ( λ ) = { λ i ∈ [ λ min + l ∆; λ min + ( l + 1)∆] }N , ∀ λ ∈ [ λ min + l ∆; λ min + ( l + 1)∆] , (7)where λ i is an eigenvalue, and l is the index of the segment. Since the maximal eigenvalue is bounded for both typesof trees (see [18]) | λ treemax | ≤ (cid:112) p − , (8)we only scan the support (cid:104) −| λ treemax | ; | λ treemax | (cid:105) . Finding the value of ∆ that provides for the most representative histogramis a separate technical question which is not addressed here. With such histograms (for both full binary trees and stars)for various levels one can easily reconstruct the desired spectral density by performing convolutions of the functions ρ k ( λ ) and g k,p ( λ ) with the distribution functions as prescribed by (3) and (6). The results of our computations of (a) Spectral density of the full binary tree T . (b) Spectral density of the full binary tree T in the log − log coordinates (to be comparedto the Fig.2b from [16]). (c) Spectral density of the exponentialensemble with parameter µ = 0 . Figure 3: Eigenvalue distribution of full binary trees. Axis X is eigenvalue, axis Y is its frequency.spectral densities (i.e. the densities of eigenvalues) for individual tree and for an ensemble of exponentially weightedtrees are summarized in Fig. 3.We would like to pay attention to the Fig. 3b which represents the same histogram as in Fig. 3a replotted in thelog − log coordinates. This figure should be compared to Fig.2b of the work [16] where the relaxational spectrum ofa specific dendrimer has been computed. The similarity between the spectral densities of: i) individual dendrimers,ii) ensembles of dendrimers, and iii) linear chains with exponential distribution in lengths, allows us to conclude thatthe spectral density does not reflect the topology of the system under consideration, demonstrating rather genericfeatures of one-dimensional systems with quenched disorder. A. Full binary trees: numerics
The number of vertices of a full binary tree grows exponentially with the level, while the structure of the spectrumis very stable, which is not surprising since the full binary tree is a self-similar structure. Note that the maximalvertex degree of a full binary tree is p = 3, so, according to (8), | λ treemax | ≤ √ k .The Fig. 3 shows typical spectral densities, namely: the Fig. 3a depicts the spectral density of a single full binary treewith k = 10 levels, and the Fig. 3c demonstrates the spectral density of the full binary tree ensemble (averaged over1000 realizations). In simulations we have generated k with the exponential distribution (2), calculated eigenvaluesof the corresponding particular tree and added all values from different trees to the list, thus considering a forest offull binary trees with prescribed level distribution. Note the similarity between Fig. 3a and Fig. 3c.For better clarity we provide numeric data for main peak multiplicities ( k = 8 , ,
10) in the Table Ia. The peaks areenumerated as shown in Fig. 3. Note that the diagonals in this table are relatively stable. In the Table Ib we presentthe main peak frequencies for k = 8 , ,
10 and also for exponentially distributed ensembles with µ = 0 . µ = 0 . Peak No. k = 8 k = 9 k = 100 85 171 3411 37 73 1462 17 34 683 8 17 334 4 8 16 (a) Peak No. µ = 0 . µ = 0 . k = 8 k = 9 k = 100 0 .
346 0 .
392 0 .
333 0 .
335 0 . .
146 0 .
152 0 .
145 0 .
143 0 . .
067 0 .
062 0 .
067 0 .
067 0 . .
032 0 .
029 0 .
031 0 .
033 0 . .
015 0 .
012 0 .
016 0 .
016 0 . (b) Table I: (a) Individual trees: peak multiplicities for various k in the enveloping series; (b) Ensemble of trees with µ = 0 . µ = 0 . k = 8 , , B. Full binary trees: theory
1. Spectral density of a single full binary tree
Let T k is a full binary tree and T k is its adjacency matrix. If the vertices are enumerated linearly through levels 1to k , T k takes the following form: T k = (9)Here k = 3, and for other k the structure is the same. However, we will not operate with this form as it does notprovide for clear analysis. Instead, we introduce alternative notation.Following [19, 20] denote B k a generalized Bethe tree of k levels, shown in Fig. 4. L e v e l D e g r ee a ndnu m b e r o f n o d e s k d k-1 d k-j+1 d n k n k-1 n k-j+1 n Figure 4: A generalized Bethe tree.
Definition 1.
A generalized Bethe tree B k is a rooted non-weighted and non-directed tree with vertex degree de-pending on the distance from the root only. All vertices in the same level have equal degree. The number of levelsequals k .Enumerate the levels down-up by index j = 1 , ..., k . The root is located at j = 1 and the number of vertices at thelevel j is n k − j +1 , all of them of equal degree d k − j +1 . In particular, d k is the degree of the root, n k = 1 and n is thenumber of terminal vertices (“leaves”).Let A ( B k ) be the adjacency matrix of B k and σ ( A ( B k )) – its spectrum. Define also d = (1 , d , . . . , d k ) andΩ = { j : 1 ≤ j ≤ k − , n j > n j +1 } . The following theorem (the Theorem 1 of [20]) is the key ingredient of ourstudy: Theorem 1. If A j ( d ) is the j × j leading principal submatrix of the k × k symmetric tridiagonal matrix A k ( d ) = √ d − √ d − √ d − √ d − . . . . . . . . . . . . . . . . . . √ d k − √ d k − √ d k √ d k (10)then1. σ ( A ( B k )) = (cid:16)(cid:83) j ∈ Ω σ ( A j ( d )) (cid:17) ∪ σ ( A k ( d )).2. The multiplicity of each eigenvalue of the matrix A j ( d ) as an eigenvalue of A ( B k ) is ( n j − n j +1 ) for j ∈ Ω, andeigenvalues of A k ( d ) as eigenvalues of A ( B k ) are simple.This theorem allows us to investigate a simple tridiagonal matrix instead of the full adjacency matrix of a tree.The size of corresponding tridiagonal matrix equals to the number of levels, which is exponentially smaller than thenumber of vertices. Now we formulate and prove the theorem about the spectrum of trees under consideration. Theorem 2.
Let T k be the full binary tree of k levels, σ ( T k ) - its spectrum, M k ( λ ) and ρ k ( λ ) are the multiplicityand the frequency of the eigenvalue λ ∈ σ ( T k ), then:1. σ ( T k ) = (cid:83) kj =1 (cid:83) ji =1 (cid:110) √ πij +1 (cid:111) .
2. M k ( λ ) = k − k mod ( j +1) j +1 − + { k ≡ j mod ( j + 1) } . ρ k ( λ ) = k − (cid:16) k − k mod ( j +1) j +1 − + { k ≡ j mod ( j + 1) } (cid:17) . Proof.
The graph T k has n k − j +1 = 2 j − vertices at the level j , all of degree d j = 3 except for the level 1 ( d = 1) andthe root ( d k = 2). According to theorem 1, its spectrum is the union of submatrix spectra: σ ( T k ) = k (cid:91) j =1 σ ( A j ) , (11)where A j is the following: A j = √ · · · · · · · · · · · · · · · (12)Let ˆ A j = 1 √ A j and ˆ λ = 1 √ λ . Note that ˆ A j is actually the adjacency matrix of the linear chain. Thecharacteristic polynomial F j = det( ˆ A j + ˆ λ E ) of the matrix ˆ A j satisfies the following recurrence equation: (cid:40) F j +1 = ˆ λF j − F j − ,F = 1 , F = ˆ λ. (13)and the corresponding characteristic equation µ − µ ˆ λ + 1 = 0 has roots µ ± = ˆ λ ± (cid:112) ˆ λ − . (14)This allows us to reconstruct the explicit form of F j : F j = 1 (cid:112) ˆ λ − µ j +1+ − (cid:112) ˆ λ − µ j +1 − . (15)Assuming ˆ λ ∈ ( −
2; 2), we solve F j = 0. Its roots are:ˆ λ = 2 cos πij + 1 , i = 1 . . . j. (16)There are j of them, and as F j is a polynomial of degree j , there are no other roots. Thus, the eigenvalues of thematrix A j are λ = 2 √ πij + 1 , i = 1 . . . j, (17)Collecting all intermediate results, we arrive at the following explicit expression for the spectrum of an individualtree T k : σ ( T k ) = k (cid:91) j =1 j (cid:91) i =1 (cid:26) √ πij + 1 (cid:27) . (18)Note that the spectrum of an individual tree is the same as the scaled spectrum of a set of linear chains, and leveldefines here the maximal length of a linear chain.Next, compute the multiplicities. According to the second statement of theorem 1, the contribution of the j th principal submatrix is: m j = n j − n j +1 = 2 k − j − ( j = 1 . . . k − , m k = 1 . (19)Recall that the multiplicity of the eigenvalue λ is M k ( λ ). Then we have:M k ( λ ) = k (cid:88) j =1 m j { λ ∈ σ ( A j ) } , (20)where {·} is the indicator function: {·} = (cid:40) , if the condition is true0 , otherwiseEigenvalue λ belongs to σ ( A j ) if there exists an integer i such that λ = 2 √ πij + 1 . (21)Supposing that ij +1 is irreducible, it contributes to the spectrum as an eigenvalue of every ( j +1) th principal submatrix:2 √ πij + 1 , √ πi j + 1) , . . . , √ n − πi ( n − j + 1) , √ nπin ( j + 1) (22)where n = (cid:106) k +1 j +1 (cid:107) . The respective multiplicities are:2 k − ( j +1) , k − j +1) , . . . , k − ( n − j +1) , m n (23)where m n depends on k and j . It is also important if λ is an eigenvalue of the matrix itself as in this case themultiplicity is 1, otherwise it is a difference of two powers of two. m n = (cid:40) , if k ≡ j mod ( j + 1)2 k − n ( j +1) = 2 k mod ( j +1) , otherwise (24)Summing the geometric series, we arrive at the following expression for the multiplicity of λ = 2 √ πij + 1 , definedin (20): M k ( λ ) = k + 2 j − j +1 − , if k ≡ j mod ( j + 1)2 k − k mod ( j +1) j +1 − , otherwise (25)or M k ( λ ) = 2 k − k mod ( j +1) j +1 − { k ≡ j mod ( j + 1) } . (26)The spectral density, ρ k , of an individual tree is ρ k ( λ ) = M k ( λ ) N tree k = 12 k − (cid:18) k − k mod ( j +1) j +1 − { k ≡ j mod ( j + 1) } (cid:19) . (27)That finishes the proof of this theorem.Now we know exactly all eigenvalues and their multiplicities or frequencies shown in Fig. 3. An important remarkis that an individual full binary tree possesses the spectral density, which is identical to the spectral density of theexponentially weighted ensemble of linear chains up to the scaling factor (the locations of the peaks are multiplied by √ ρ outk ( λ ) as the series of main peaks, sequentially enumerated as shown in Fig. 3. It is alsothe series of maximal eigenvalues of levels from 2 to k : λ outj = 2 √ πj + 1 , j = 2 , . . . , k. (28)The second series defines the “inner curve” ρ ink ( λ ), which is the enveloping curve of the peaks descending from thesecond main peak to zero. In other words, it is the series of minimal positive eigenvalues: λ ini = 2 √ πj j + 1 = 2 √ π j + 1) , j = 1 , . . . , (cid:22) k (cid:23) . (29) Theorem 3.
Let T k be the full binary tree of level k , then the inner curve ρ ink ( λ ) and the outer curve ρ outk ( λ ) of itsspectral density ρ k ( λ ) have exponential asymptotics:1. ρ outk ( λ ) ∼ exp (cid:16) − A √ λ max − λ (cid:17) at λ → λ max ,2. ρ ink ( λ ) ∼ exp (cid:0) − Bλ (cid:1) at λ → A and B are some positive absolute constants. Proof.
Considering the asymptotics of the tail of the “outer curve” , or its behavior as λ → λ max = 2 √
2, is the sameas considering k (cid:29) j (cid:29) λ = 2 √ πj + 1 = 2 √ (cid:18) − π j + 1) (cid:19) + o (cid:18) j + 1) (cid:19) (30)Inverting (30), we get j + 1 = √ π (cid:112) √ − λ + o (cid:18) j + 1) (cid:19) (31)Both i and λ are in fact discrete and this expression only provides for a good approximation. Substituting (31) into(27), we obtain the asymptotic expression of the tail ρ outk ( λ ) near the spectrum edge, i.e. at | λ | → λ max : ρ outk ( λ ) = exp − √ π ln 2 (cid:113) √ − | λ | (1 + o (1)) ∼ exp (cid:32) − A (cid:112) λ max − | λ | (cid:33) , A = √ π ln 2 . (32)Now consider the asymptotics of the inner curve at j (cid:29) k (cid:29) λ → λ = 2 √ π j + 1) = π √ j + 1) + o (cid:18) j + 1 (cid:19) ; (33)Inverting this expression and substituting it into (27) in the same manner we did for the outer curve, we arrive atthe following asymptotics of the inner curve (at λ → ρ ink ( λ ) ∼ exp (cid:32) − √ π ln 2 | λ | (cid:33) = exp (cid:18) − B | λ | (cid:19) , B = √ π ln 2 . (34)The behavior (32) signals the existence of so-called “Lifshitz tail” typical for the edge of the spectral density inphysics of one-dimensional disordered systems [5, 23]. The emergence of similar behavior in deterministic tree-likesystem allows to suggest that regular tree-like graphs could be served as models which mimic some properties ofensembles of one-dimensional disordered systems like, for example, the one-dimensional disordered alloys (see [24] forpresenting the issue and a number of related references).
2. Spectral density of full binary tree ensembles
In this section we derive the eigenvalues and their frequencies for the ensemble of full binary trees, which is a setof full binary trees with sizes distributed exponentially. It turns out that, indeed, the results are pretty similar to thecase of a single tree.
Theorem 4.
Let σ ensemble be the spectrum of the full binary tree ensemble with level distributed exponentially, and ρ ( λ ) be the spectral density of such ensemble, then1. σ ensemble = (cid:83) ∞ j =1 (cid:83) ji =1 (cid:110) √ πij +1 (cid:111) ,
2. For every λ ∈ σ ensemble , i.e. λ = 2 √ πij +1 , where i and j are coprime, ρ ( λ ) = ( e µ − e µ ( j +1) − µ > ln 2, and ρ ( λ ) = j +1 − otherwise.1 Proof.
The first statement is obvious. The spectrum of the binary tree ensemble σ ensemble is the union of spectra withevery possible k . It reads σ ensemble = ∞ (cid:91) k =1 σ ( T k ) = ∞ (cid:91) k =1 k (cid:91) j =1 (cid:26) √ πjk + 1 (cid:27) , (35)where T k is the full binary tree of level k . Compare to (18), note that k → ∞ instead of being finite.Recall that level k is distributed exponentially, or P µ ( k ) = ( e µ − e − µk . The spectral density ρ ( λ ) of the ensembleis the following quotient: ρ ( λ ) = E µ M k ( λ ) E µ N k = ∞ (cid:80) k =1 M k ( λ ) e − µk ∞ (cid:80) k =1 (2 k − e − µk (36)Note that both series do not always converge. This fact is due to number of vertices, or tree size, growing exponentiallywith level. Substituting here the expression for multiplicities, we get: ρ (cid:18) √ πij + 1 (cid:19) = 12 j +1 − ∞ (cid:80) k = j +1 (2 k − k mod ( j +1) ) e − µk + ∞ (cid:80) k = j e − µk { k ≡ j mod ( j + 1) } ∞ (cid:80) k =1 (2 k − e − µk (37)For µ > ln 2 ≈ .
69 all series converge. Computing the corresponding geometric sums, we obtain ρ (cid:18) √ πij + 1 (cid:19) = ( e µ − e µ ( j +1) − µ < ln 2 the sums diverge, however the limit of their partial sums quotient exists, does not depend on µ and isequal to ρ (cid:18) √ πij + 1 (cid:19) = 12 j +1 − µ → ln 2. Another important remark is that these expressionsare indeed very similar to the individual case, and this resemblance is especially evident for µ < ln 2.Our analytic results are perfectly consistent with the numeric simulations. Theoretical and practical values areavailable in tables IIa and IIb. Peak No. µ = 0 . µ = 0 . µ = 0 . µ = 1 .
00 0 . . . . . . . . . . . . . . . . . . . . (a) Theoretical peak frequencies for µ = 0 . , . , . .
0. The first two columnsare calculated from (39), other two - from (38).
Peak No. µ = 0 . µ = 0 . µ = 0 . µ = 1 .
00 0 . . . . . . . . . . . . . . . . . . . . (b) Peak frequencies for the same ensemblesaccording to the simulation. Table II: Comparison of the theoretical and experimental (computer simulation) values of main peak frequencies inspectral densities of full binary tree ensembles with level distributed exponentially with various µ .We are in position to describe the envelope of the spectral density in the ensemble. In this case we again painattention to “outer” and “inner” curves. The outer curve ρ out ( λ ) is defined by the series: λ outj = 2 √ πj + 1 , j = 2 , , , . . . (40)2while the inner curve ρ in ( λ ) is set by: λ inj = 2 √ π j (2 j + 1) , j = 2 , , , . . . . (41)Let us formulate the theorem regarding the asymptotics of the inner and outer curves. Theorem 5.
Let ρ ( λ ) be the spectral density of the full binary tree ensemble with level k distributed exponentiallywith parameter µ , then the outer curve ρ out ( λ ) and the inner curve ρ in ( λ ) have the following asymptotics:1. ρ out ( λ ) ∼ A µ exp (cid:18) − B µ √ λ max − λ (cid:19) at λ → λ max ,2. ρ in ( λ ) ∼ A µ exp (cid:18) − C µ λ (cid:19) at λ → A µ , B µ , C µ are some positive constants independent of λ . Proof.
First, consider µ > ln 2. Following the proof of Theorem 3, use the Taylor expansion of λ outj at j (cid:29) λ = 2 √ πj + 1 = 2 √ (cid:18) − π j + 1) (cid:19) + o (cid:18) j + 1) (cid:19) . (42)Inverting this expression, we get j + 1 = √ π (cid:112) √ − λ + o (cid:18) j + 1) (cid:19) (43)and substitute it into the spectral density (38) we obtained in the previous theorem: ρ ( λ ) ∼ ( e µ − exp (cid:32) − √ πµ √ λ max − λ (cid:33) . (44)As for the inner curve, again, use the Taylor expansion of λ inj at j (cid:29) λ = 2 √ π j + 1) = π √ j + 1) + o (cid:18) j + 1 (cid:19) , (45)then invert it, getting 2 j + 1 = π √ λ + o (cid:18) j + 1 (cid:19) . (46)This expression then is substituted into (38) and the final result is ρ ( λ ) ∼ ( e µ − exp (cid:32) √ πµλ (cid:33) . (47)Collecting these two formulas together, we get the shape of ρ ( λ ): ρ ( λ ) ∼ A µ exp (cid:18) − B µ √ λ max − λ (cid:19) , for λ → λ max A µ exp (cid:18) − C µ λ (cid:19) , for λ → A µ = ( e µ − , B µ = √ πµ and C µ = √ πµ . Recall that the density in the case of µ < ln 2 is simply thesame as if we just substituted µ = ln 2 in the case of µ > ln 2. It means that to obtain the expressions of the innerand outer curves we also have to assume µ = ln 2 in (48). By doing that we get exactly the same expressions as (32)and (34), corresponding to A µ = 1 , B µ = √ π ln 2 and C µ = √ π ln 2.3To sum it up, we have evaluated the spectra of a single full binary tree and of the exponentially weighted ensembleof full binary trees by computing explicitly all multiplicities (frequencies). We found ρ ( λ ) at the edges of the spectrum,known as the “Lifshitz tail” [5, 23], typical for the Anderson localization in disordered one-dimensional systems suchas the one-dimensional Schr¨odinger-like discrete operators with a diagonal disorder. Two interesting peculiarities ofconsidered systems are worth to mention: i) even a single large tree has the Lifshitz tail; ii) the spectral density of theexponentially weighted ensemble of full binary trees has the same asymptotic behavior at the edge of the spectrumas the exponential ensemble of linear chains. C. Star graphs: numerics
We begin the analysis of the star-like graphs by calculating their spectra with different parameters k and p (seeFig. 2b). In particular, we consider k = 1 , . . . ,
500 and p = 3 , . . . ,
10, which is possible as the size of a star is linearin terms of both level k and root degree p . We emphasize that the spectrum and spectral density of an individualstar do depend on k and p , but their structure is fixed. The spectrum always consists of two separate series: k − p − k simple eigenvalues as one can observe in Fig. 5. Note that almost all eigenvaluesbelong to the segment [ −
2; 2], and there are also two symmetric with respect to zero outstanding eigenvalues. (a) Spectral density of S , (b) Spectral density of S , Figure 5: Eigenvalue distribution for stars. Axis X is eigenvalue, axis Y is its frequency.Unlike the case of full binary trees, the spectral density of the exponential star ensemble is essentially different fromthe individual star spectral density. What is more, the structure of this spectral density depends on µ notedly asshown in Fig. 8. This behavior should be compared to the one considered in [25] for tree-like graphs with a “heavy”root, where the localization-delocalization of the spectrum has been found. D. Star graphs: theory
1. Spectrum of a single star graph
Here we derive the spectrum of an individual star graph. We pay special attention of computing multiplicities(frequencies) of eigenvalues.
Theorem 6.
Let S k,p be the star of k levels and root degree p , σ ( S k,p ) - its spectrum and g k,p ( λ ) - its spectraldensity, then:41. σ ( S k,p ) = σ link ∪ σ ck,p , where σ link = k − (cid:83) j =1 (cid:26) πjk (cid:27) and σ ck,p is the set of root of the complex equationtan kϕ = pp − tan ϕ, λ = 2 cos ϕ .2. g k,p ( λ ) = p − p ( k −
1) + 1 , if λ ∈ σ link , p ( k −
1) + 1 , if λ ∈ σ ck,p Proof.
For the star S k,p of p branches and k levels, shown in Fig. 2b, according to Theorem 1 [20], the spectrum isthe set of eigenvalues of j × j principal submatrices ( j = 1 . . . k ). A ( S k,p ) = . . . . . . . . . . √ p √ p (49)First, Theorem 1 suggest constructing the set Ω of indices such that n j > n j +1 as the proposed multiplicities are m j = n j − n j +1 , j = 1 . . . k − m k = 1 . (50)We notice that this set consists of one index only as for star graphs n j = p for j = 1 , . . . , k −
1, and n k = 1. Hence, m j = 0 for j = 1 , . . . , k −
2, and we only need to compute the spectra of A k − (eigenvalues of which have multiplicity p −
1) and A k (eigenvalues of which are simple). These are the two separate series we pointed out in the numericalsection.This leads us to the following conjecture: we split the spectrum of a star into two parts. The first one is the linearchain contribution σ link as the spectrum of A k − , and the second one in the center contribution σ ck,p as the spectrumof A k . As the total number of eigenvalues equals p ( k −
1) + 1, the final expression for the spectral density as in (4) is g k,p ( λ ) = M k,p ( λ ) p ( k −
1) + 1 = p − p ( k −
1) + 1 , if λ ∈ σ link , p ( k −
1) + 1 , if λ ∈ σ ck,p (51)The roots of F k − , derived in (15), define the spectrum σ link : σ link = k − (cid:91) j =1 (cid:26) πjk (cid:27) (52)Computing σ ck,p is a much more difficult task. Applying the Laplace formula to the last row of matrix A k twotimes, we get: G k = det A k = λF k − − pF k − = pµ + − λ √ λ − µ k − − pµ − − λ √ λ − µ k + (53)where µ ± = λ ± √ λ −
42 (54)The equation G k = 0 is equivalent to the following complex-valued transcendental equation:tan kϕ = pp − ϕ, (55)5where we have introduced ϕ as follows: tan ϕ = √ − λ λ , ( λ = 2 cos ϕ ) . (56)The roots of this equation form σ cn,p . This set is always unique, except for the roots of type λ = 2 cos πij , where i and j are coprime, or in other words, for any n (cid:54) = l :( σ cn,p ∩ σ cl,p ) \ σ lin = ∅ (57)And also for any n σ cn,p ∩ σ linn = ∅ (58) Theorem 7.
Let S k,p be the star with p branches and k levels, σ ( S k,p ) be its spectrum and σ ck,p be the centralcontribution in σ ( S k,p ), then1. the maximal eigenvalue in the whole spectrum belongs to σ ck,p and lies in the interval ( √ p ; p √ p − ).2. the rest of the values from σ ck,p belong to the interval ( −
2; 2) and each positive value localizes in the interval λ t ∈ (cid:16) (cid:16) πtk + k arctan (cid:16) pp − tan πtk (cid:17)(cid:17) ; 2 cos (cid:0) πtk + π k (cid:1)(cid:17) , t = 1 , , . . . , (cid:2) k − (cid:3) , with a corresponding negativevalue of the same absolute value. There is also λ = 0 for any odd k . Proof.
Recall that σ ck,p is the set of roots of the complex-valued equationtan kϕ = pp − ϕ ( λ = 2 cos ϕ ) . (59)The maximal eigenvalue λ maxk is always the eigenvalue of A k , so it always belongs to σ ck,p .Despite this equation cannot be solved exactly, a transparent analysis of its solution is available. Here is a briefexplanation of how we localize the maximal root, λ maxk for various p and k . Computing G k as defined in (53) forsmall k , we get ( p ≥ G = λ − p ⇒ λ max = √ p,G = λ ( λ − ( p + 1)) ⇒ λ max = √ p + 1 ... (60)For λ ≥ ϕ = √ λ − λ ( λ = 2 cosh ϕ ) . (61)For ϕ > y k ( ϕ ) = tanh kϕ and z ( ϕ ) = pp − tanh ϕ are both monotone and concave. The illustration of this can be seen in Fig. 6. If y (cid:48) k (0) ≤ z (cid:48) (0),there are no roots, and there is one otherwise, which happens when k > pp − .Note that p ≥
3, so pp − ≤
3. Also note that if k > l , y k > y l for any ϕ >
0. As a result, λ maxk is strictly monotonicand λ max = √ p is the minimal value in this series. For any k the function y k ( ϕ ) <
1, thus the upper bound for thesolution is: pp − ϕ = 1 ⇒ λ = p √ p − λ max k ∈ ( √ p, p √ p − ). Now we are moving on to the second statement of the theorem.6 f ( ϕ ) pp−2 tanh(ϕ)tanh(3ϕ)tanh(5ϕ)tanh(10ϕ) (a) Plots for p = 3 f ( ϕ ) pp−2 tanh(ϕ)tanh(3ϕ)tanh(5ϕ)tanh(10ϕ) (b) Plots for p = 4 Figure 6: Illustration of graphic solution of (55).For λ ∈ ( −
2; 2) the equation (56) transforms into the real-valued equation in the same form, but with real ϕ tan kϕ = pp − ϕ. (63)First, let us count its roots. It is clear that we should only count the roots in the segment [0; π ] as the number ofnegative roots is the same as of positive ones and due to both functions being periodic, all other values would givethe same λ = 2 cos ϕ .The period of f k ( ϕ ) = tan kϕ is πk , the period of f ( ϕ ) = pp − tan ϕ is π . There are no non-zero roots in thesegment [0; π k ] as the derivative of f k is always larger than that of f . Zero is excluded from the possible roots bythe denominator in G k . In every other period of f k , the segment [ π k + πtk ; π k + π ( t +1) k ] with t = 1 , , . . . , (cid:2) k (cid:3) , there isstrictly one root, as f k is unbounded and f is bounded and both are convex and monotone.If k is even, the last period of f k is actually a semi-period, so there are no roots there belonging to [0; π ]. Thusthere are k − k − − π ; π ] and gives us k − λ . If k is odd,we consider ϕ = π a solution as well, which corresponds to λ = 0. So, this gives us k − · − λ , whichis again k −
2. This idea is illustrated by Fig. 7. Recall that there were two more values. So, that makes k rootsaltogether, which is exactly the number of roots for a polynomial of degree k . This means that we have found all theroots: k − −
2; 2) and two more in ( − p √ p − ; −√ p ) ∪ ( √ p, p √ p − ).Now consider k → + ∞ . Recall that we indexed the periods of f k with t . For small t every period f is almostconstant. This means that the solution of f ( ϕ ) = f k ( ϕ ) approximately suffices f k ( ϕ ) = tan kϕ = pp − πtk ≈ f (cid:18) πtk (cid:19) . (64)This would actually give the lower bound on the t -th root ϕ t . So, the approximate solution lies in the interval ϕ t ∈ (cid:18) πtk + 1 k arctan (cid:18) pp − πtk (cid:19) ; πtk + π k (cid:19) . (65)For bigger t the root tends to the right end of the segment, which is π k + πtk . The corresponding λ then lies in thefollowing interval: λ t ∈ (cid:18) (cid:18) πtk + 1 k arctan (cid:18) pp − πtk (cid:19)(cid:19) ; 2 cos (cid:18) πtk + π k (cid:19)(cid:19) , (66)where t is the index of the positive root counting from the right edge of the spectrum.Recall Fig. 5 and notice that, indeed the values from σ ck,p tend to the values from σ linp near the edge.7 (a) Plots for p = 3 , , k = 10. (b) Plots for p = 3 , , k = 15. Figure 7: Illustration of the root localization for the equation tan kϕ = pp − tan ϕ
2. Spectral density of ensemble of star graphs
In this section we derive the spectral density of the star graph ensemble. We also discuss its connection to thelinear chain ensemble. Here we again discriminate the two parts of the spectrum: σ ensemblep = σ lin ∪ σ cp , (67)where p is the degree of the root, and σ lin = + ∞ (cid:91) k =2 σ link , σ cp = + ∞ (cid:91) k =1 σ ck,p . (68)Note that, formally, σ ( S ,p ) = σ c ,p and σ lin = ∅ . For this purpose we also divide the spectral density into the sumof contributions from the “arms” (linear chain contribution) and the “root” (center contribution), g p ( λ ) = g linp ( λ ) + g cp ( λ ) (69)and formulation the following theorem using this notation. Theorem 8.
Let σ ensemblep be the spectrum of the star ensemble with level distributed exponentially with parameter µ and g p ( λ ) = g linp ( λ ) + g cp ( λ ) be the spectral density of such ensemble, then1. σ ensemblep = (cid:83) ∞ k =2 (cid:83) k − j =1 (cid:8) πjk (cid:9) ∪ (cid:83) ∞ k =1 σ ck,p ,2. g linp ( λ ) = ( p − e µ − ( e µj − e µ + p − ,3. g cp ( λ ) = e − µk ( e µ − e µ + p − . Proof.
The first statement is trivial: σ lin = ∞ (cid:91) k =2 σ link = ∞ (cid:91) k =2 k − (cid:91) j =1 (cid:26) πjk (cid:27) , σ cp = ∞ (cid:91) k =1 σ ck,p . (70)8Any λ ∈ σ lin can be written in the form λ = 2 cos πij , where i and j ∈ N and are coprime. Such λ belongs to σ lintj for any t ∈ N with the multiplicity p −
1. Then the spectral density g linp ( λ ) reads: g linp ( λ ) = ∞ (cid:80) k =1 e − µk ( p − { λ ∈ σ link } ∞ (cid:80) k =1 ( p ( k −
1) + 1) e − µk = ( p − ∞ (cid:80) k =1 e − µjk ∞ (cid:80) k =1 ( p ( k −
1) + 1) e − µk (71)Both series, in the nominator and in the denominator of (71), converge for any µ >
0. Evaluating these series, we get: g linp ( λ ) = ( p − e µ − ( e µj − e µ + p −
1) (72)The function g linp ( λ ) is the rescaled spectral density of the linear chain ensemble, derived in [4]: g linp ( λ ) = ( p − e − µ p − e − µ ρ lin ( λ ) = ( p − e − µ p − e − µ lim N →∞ ε → επN N (cid:88) n =1 µ n n (cid:88) k =1 λ − πkn +1 ) + ε (73)Hence, g linp ( λ ) shares all the properties of ρ lin ( λ ).For a unique eigenvalue λ from σ cp , such that λ ∈ σ ck,p one has: g cp ( λ ) = e − µk ∞ (cid:80) k =1 ( p ( k −
1) + 1) e − µk = e − µk ( e µ − e µ + p − σ cp are unique as they generally do not satisfy λ = 2 cos πij . We should also point out that λ = 0 belongs to σ ck,p for any odd k and to σ link for any even k : g cp (0) = ∞ (cid:80) k =1 e − µ (2 k − ∞ (cid:80) k =1 ( p ( k −
1) + 1) e − µk = ( e µ − ( e µ − p − e − µ ) (75)and g linp (0) = ( p − e µ − ( e µ − e µ + p −
1) (76)Thus, we get that frequency (degeneracy) of the central peak does not depend on p : g p (0) = g linp (0) + g cp (0) = e µ − e µ + 1 (77)This result is perfectly consistent with numeric simulations. Another notable result is that (73) shows that theconnection we assumed does exist. This effect is illustrated by Fig. 8.We should point out that there is no Lifshitz tail for the enveloping curve like the one found in (48). However, stillfor ensemble of stars we have a kind of a Lifshitz tail in the main part of the spectrum near λ = 2, just the same asfor linear chains [4]. The “hand-waving” explanation is as follows: the height of the linear part of the spectrum is p − πk occurs in every k -th level of the star, whereas the values from the centralpart are mostly unique.9 −2 −1 0 1 2λ0.0000.0050.0100.0150.0200.0250.0300.0350.0400.045 g ( λ ) (a) µ = 0 . −2 −1 0 1 2λ0.000.050.100.150.200.25 g ( λ ) (b) µ = 0 . −2 −1 0 1 2λ0.00.10.20.30.4 g ( λ ) (c) µ = 1 . Figure 8: Eigenvalue distribution of star-like graphs with root branching p = 3 in exponential ensembles withvarious µ . Axis X is eigenvalue, axis Y is its frequency. IV. CONCLUSION
In this work we have investigated the spectral statistics of exponentially weighted ensembles of full binary treesand p -branching star graphs. We have shown that corresponding spectral densities demonstrate peculiar ultrametricstructure, typical for sparse graphs and for ensembles of linear chains exponentially distributed in lengths.Both models, presented in the work, the fully branching 3-valent trees and the star-like graphs, have interestingpeculiarities which are worth mentioning: • The spectral density of a single tree already shares the ultrametric behavior, which is changed only by scalingfactor when passing from a single tree to the ensemble of trees exponentially distributed in their sizes. • The spectral density of the exponential ensemble of trees looks very similar to the one of linear chains, sharingthe same asymptotic behavior of tails of the distribution, ∼ e − c/ √ λ max − λ , where for p -branching trees one has λ max = 2 √ p − p = 2 and λ linmax = 2, while for 3-branching trees p = 3 and λ treemax = 2 √ • The spectral statistics of p -branching star-like graphs strongly depends on p (the branching of the root point)and on µ (the parameter in the exponential distribution of graphs sizes).The edge singularity ∼ e − c/ √ λ max − λ at λ → λ max reproduces the corresponding Lifshitz tail of the one-dimensionalAnderson localization in a random Schr¨odinger operator [5, 23, 26] with strong disorder: ρ ( E ) ∼ e − /E d/ where E = λ max − λ and d = 1. The appearance of the edge singularity ρ ( E ) ∼ e − / √ E in our situation is purelygeometric, it does not rely on any entropy-energy-balance consideration like optimal fluctuation [5, 23]. Moreover, weshould emphasize an universality of a Lifshitz tail: typically, Anderson localization appears for random three-diagonaloperators with the randomness on the main diagonal, while in our case there is no diagonal disorder and the edgesingularity has a kind of a number theoretic signature.In a more practical setting, our analysis demonstrates that measuring relaxational times of dilute solutions ofdendrimers, or of linear chains exponentially distributed in lengths, or of sparse clusters, we arrive in all that cases atvery similar hierarchically organized spectral densities. So, our study can be regarded as a warning for chemists: therelaxational spectrum is not determined by the cluster topology, but has rather the number-theoretic origin, reflectingthe peculiarities of the rare-event statistics typical for one-dimensional systems with a quenched structural disorder.Very promising looks the similarity of spectral density of individual dendrimer and of ensemble of linear chains withexponential distribution in lengths. The fact that dendrimers mimic some properties of ensembles of one-dimensionaldisordered systems signifies that dendrimers could be served as simple disorder-less toy models of one-dimensionalsystems with quenched disorder.Investigation of spectral properties of star-like graphs with different branchings should be compared to the results ofthe work [25], where two different techniques have been considered simultaneously: the study of spectral properties ofthe graph adjacency matrix for finite graphs, and the study of singularities of the grand canonical partition function.The information acquired from the spectral density of the graph (or of ensemble of graphs) can be straightforwardlytranslated to the diffusion of point-like excitations on the graph (on ensemble of graphs). The interplay between0the length of arms and the branching number can lead to the localization of such excitations due to purely entropicreasons. Acknowledgments
The authors are very grateful to M. Tamm and K. Polovnikov for numerous discussions and valuable comments. SNacknowledges the support by RFBR grant 16-02-00252 and by EU-Horizon 2020 IRSES project DIONICOS (612707).The work of VK and YM at Skolkovo Institute of Science and Technology is supported in part by the grant of thePresident of Russian Federation for young PhD MK-9662.2016.9 and by the RFBR grant 15-07-09121a. The work atLANL was carried out under the auspices of the National Nuclear Security Administration of the U.S. Departmentof Energy under Contract No. DE-AC52-06NA25396. [1] A. E. Brouwer and W. H. Haemers,
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