Bethe Trees Research Articles

The largest ABC eigenvalue and the ABC energy of generalized Bethe trees

The largest ABC eigenvalue and the ABC energy of generalized Bethe trees

Degree-weighted Wiener index of a graph

<abstract><p>From geometric point of view, we introduced the Sombor-Wiener index of a graph and studied the basic properties of the new index. It was shown that the Sombor-Wiener index was useful in predicting the acentric factor of octane isomers. In addition, we proposed a degree-weighted Wiener index to generalize the Schultz index, the Gutman index, and the Sombor-Wiener index. Meanwhile, we gave the calculation formula of degree-weighted Wiener index for generalized Bethe trees.</p></abstract>

Disproof of a conjecture on the main spectrum of generalized Bethe trees

An eigenvalue of the adjacency matrix of a graph is said to be main if the all-ones vector is not orthogonal to its associated eigenspace. A generalized Bethe tree with k levels is a rooted tree in which vertices at the same level have the same degree. França and Brondani (2021) [4] recently conjectured that any generalized Bethe tree with k levels has exactly k main eigenvalues whenever k is even. We disprove the conjecture by constructing a family of counterexamples for even integers k≥6.

The general Albertson irregularity index of graphs

<abstract><p>We introduce the general Albertson irregularity index of a connected graph $ G $ and define it as $ A_{p}(G) = (\sum_{uv\in E(G)}|d(u)-d(v)|^p)^{\frac{1}{p}} $, where $ p $ is a positive real number and $ d(v) $ is the degree of the vertex $ v $ in $ G $. The new index is not only generalization of the well-known Albertson irregularity index and $ \sigma $-index, but also it is the Minkowski norm of the degree of vertex. We present lower and upper bounds on the general Albertson irregularity index. In addition, we study the extremal value on the general Albertson irregularity index for trees of given order. Finally, we give the calculation formula of the general Albertson index of generalized Bethe trees and Kragujevac trees.</p></abstract>

On the main spectrum of generalized Bethe trees

An eigenvalue of the adjacency matrix of a graph is said to be main if the all-ones vector is not orthogonal to its associated eigenspace. In this paper, we investigate the main eigenvalues of generalized Bethe trees and determine the number of main eigenvalues of some subclasses of these trees. In particular, we characterize the main spectrum of Bethe trees.

The normalized Laplacian polynomial of rooted product of graphs

Let [Formula: see text] be a simple graph with [Formula: see text] vertices and [Formula: see text] be a sequence of [Formula: see text] rooted graphs [Formula: see text]. The rooted product [Formula: see text], of [Formula: see text] by [Formula: see text] is constructed by identifying the root vertex of [Formula: see text] with the [Formula: see text]th vertex of [Formula: see text]. In this paper, the characteristic polynomial of the normalized Laplacian matrix of [Formula: see text] is obtained. As an application of our results, we obtain the normalized Laplacian polynomial and spectrum of the generalized Bethe trees.

Multi-distance granularity structural α-subtree index of generalized Bethe trees

In graph theory and its applications, trees, BC-trees, subtrees and BC-subtrees have been extensively studied. We introduce a generalization of the BC-tree, called the multi-granular α-tree, which is a tree (of order at least α+1) where any two leaves are at a distance that is a multiple of α. We study the number of α-subtrees, through α-subtree generating functions, for generalized Bethe trees, Bethe trees and dendrimers (hyper-branched structures in molecular topology). Our results can also be used to examine the asymptotic behavior of the average order of α-subtrees in dendrimers.

Periodicity of Grover walks on generalized Bethe trees

We focus on the periodicity of the Grover walk on the generalized Bethe tree, which is a rooted tree such that in each level the vertices have the same degree. Since the Grover walk is induced by the underlying graph, its properties depend on the graph. In this paper, we say that the graph induces periodic Grover walks if and only if there exists k∈N such that the k-th power of the time evolution operator becomes the identity operator. Our aim is to characterize such graphs. We give the perfect characterizations of the generalized Bethe trees which induce periodic Grover walks.

Loop expansion around the Bethe approximation through the M-layer construction

For every physical model defined on a generic graph or factor graph, the Bethe M-layer construction allows building a different model for which the Bethe approximation is exact in the large M limit, and coincides with the original model for . The perturbative series is then expressed by a diagrammatic loop expansion in terms of so-called fat diagrams. Our motivation is to study some important second-order phase transitions that do exist on the Bethe lattice, but are either qualitatively different or absent in the corresponding fully connected case. In this case, the standard approach based on a perturbative expansion around the naive mean field theory (essentially a fully connected model) fails.On physical grounds, we expect that when the construction is applied to a lattice in finite dimension there is a small region of the external parameters, close to the Bethe critical point, where strong deviations from mean-field behavior will be observed. In this region, the expansion for the corrections diverges, and can be the starting point for determining the correct non-mean-field critical exponents using renormalization group arguments. In the end, we will show that the critical series for the generic observable can be expressed as a sum of Feynman diagrams with the same numerical prefactors of field theories. However, the contribution of a given diagram is not evaluated by associating Gaussian propagators to its lines, as in field theories: one has to consider the graph as a portion of the original lattice, replacing the internal lines with appropriate one-dimensional chains, and attaching to the internal points the appropriate number of infinite-size Bethe trees to restore the correct local connectivity of the original model.The actual contribution of each (fat) diagram is the so-called line-connected observable, which also includes contributions from sub-diagrams with appropriate prefactors. In order to compute the corrections near to the critical point, Feynman diagrams (with their symmetry factors) can be read directly from the appropriate field-theoretical literature; the computation of momentum integrals is also quite similar; the extra work consists of computing the line-connected observable of the associated fat diagram in the limit of all lines becoming infinitely long.

On the Spectra of Reduced Distance Matrix of the Generalized Bethe Trees

Let G be a simple connected graph and {v_1,v_2,..., v_k} be the set of pendent (vertices of degree one) vertices of G. The reduced distance matrix of G is a square matrix whose (i,j)-entry is the topological distance between v_i and v_j of G. In this paper, we compute the spectrum of the reduced distance matrix of the generalized Bethe trees.

On the Aα-spectra of trees

Let G be a graph with adjacency matrix A(G) and let D(G) be the diagonal matrix of the degrees of G. For every real α∈[0,1], define the matrix Aα(G) asAα(G)=αD(G)+(1−α)A(G).This paper gives several results about the Aα-matrices of trees. In particular, it is shown that if TΔ is a tree of maximal degree Δ, then the spectral radius of Aα(TΔ) satisfies the tight inequalityρ(Aα(TΔ))<αΔ+2(1−α)Δ−1, which implies previous bounds of Godsil, Lovász, and Stevanović. The proof is deduced from some new results about the Aα-matrices of Bethe trees and generalized Bethe trees.In addition, several bounds on the spectral radius of Aα of general graphs are proved, implying tight bounds for paths and Bethe trees.

Exploring Symmetries to Decompose Matrices and Graphs Preserving the Spectrum

Given a special kind of symmetric matrix, we present a decomposition technique that preserves its spectrum. We transform this result into an algorithm for graphs. The algorithm disconnects the graph, resulting in smaller matrices, reducing the complexity of computing the graph spectrum. This technique may be seen as a unified approach of several decomposition techniques present in the literature for different instances of matrices and classes of graphs. As an application, we use the algorithm for three classes of graphs: threshold graphs, generalized Bethe trees, and multifan graphs, obtaining expressions for their spectra, for various matrices.

On the smallest eigenvalues of the line graphs of some trees

In this paper, we study the characteristic polynomials of the line graphs of generalized Bethe trees. We give an infinite family of such graphs sharing the same smallest eigenvalue. Our family generalizes the family of coronas of complete graphs discovered by Cvetković and Stevanović.

Computing the Laplacian spectra of some graphs

In this paper we give a simple characterization of the Laplacian spectra of a family of graphs as the eigenvalues of symmetric tridiagonal matrices. In addition, we apply our result to obtain upper and lower bounds for the Laplacian-energy-like invariant of these graphs. The class of graphs considered are obtained from copies of modified generalized Bethe trees (obtained by joining the vertices at some level by paths), identifying their roots with the vertices of a regular graph or a path.

Line graph of combinations of generalized Bethe trees: Eigenvalues and energy

We characterize the eigenvalues and energy of the line graph L ( G ) whenever G is (i) a generalized Bethe tree, (ii) a Bethe tree, (iii) a tree defined by generalized Bethe trees attached to a path, (iv) a tree defined by generalized Bethe trees having a common root, (v) a graph defined by copies of a generalized Bethe tree attached to a cycle and (vi) a graph defined by copies of a star attached to a cycle; in this case, explicit formulas for the eigenvalues and energy of L ( G ) are derived.

Finite-size scaling of the de Almeida–Thouless instability in random sparse networks

We study, in random sparse networks, finite-size scaling of the spin-glass susceptibility ${\ensuremath{\chi}}_{\text{SG}}$, which is a proper measure of the de Almeida--Thouless (AT) instability of spin-glass systems. Using a phenomenological argument regarding the band-edge behavior of the Hessian eigenvalue distribution, we discuss how ${\ensuremath{\chi}}_{\text{SG}}$ is evaluated in infinitely large random sparse networks, which are usually identified with Bethe trees, and how it should be corrected in finite systems. In the high-temperature region, data of extensive numerical experiments are generally in good agreement with the theoretical values of ${\ensuremath{\chi}}_{\text{SG}}$ determined from the Bethe tree. In the absence of external fields, the data also show a scaling relation ${\ensuremath{\chi}}_{\text{SG}}={N}^{1/3}F({N}^{1/3}|T\ensuremath{-}{T}_{c}|/{T}_{c})$, which has been conjectured in the literature, where ${T}_{c}$ is the critical temperature. In the presence of external fields, on the other hand, the numerical data are not consistent with this scaling relation. A numerical analysis of Hessian eigenvalues implies that strong finite-size corrections of the lower band edge of the eigenvalue distribution, which seem relevant only in the presence of the fields, are a major source of inconsistency. This may be related to the known difficulty in using only numerical methods to detect the AT instability.

Applications of recurrence relations for the characteristic polynomials of Bethe trees

A Bethe tree of k levels, B k ( d ) , is a rooted tree such that the root vertex has degree d , the vertices from level 2 to k − 1 have degree d + 1 and the vertices at level k are leaves. In this paper, we obtain a recurrence relation for the characteristic polynomial and the Laplacian characteristic polynomial of Bethe trees. As an application, we prove that there are no integral Bethe trees except for the star B 2 ( n 2 ) , and we obtain a recurrence relation for the Laplacian-energy-like invariant of Bethe trees.

Improved bounds for the Laplacian energy of Bethe trees

For k , d ⩾ 2 , a Bethe tree is a rooted tree with k levels which the root vertex has degree d , the vertices from level 2 to k - 1 have degree d + 1 and the vertices at the level k are pendent vertices. So et al, using a theorem by Ky Fan have obtained both upper and lower bounds for the Laplacian energy of bipartite graphs. We shall employ the above mentioned theorem to obtain new and improved bounds for the Laplacian energy in the case of Bethe trees.

Spectra of weighted compound graphs of generalized Bethe trees

A generalized Bethe tree is a rooted tree in which vertices at the same distance from the root have the same degree. Let Gm be a connected weighted graph on m vertices. Let {Bi :1≤ i ≤ m} be a set of trees such that, for i =1 ,2,...,m , (i) Bi is a generalized Bethe tree of ki levels, (ii) the vertices of Bi at the level j have degree di,ki−j+1 for j =1 ,2,...,ki, and (iii) the edges of Bi joining the vertices at the level j with the vertices at the level (j + 1) have weight wi,ki−j for j =1 ,2,...,k i −1. Let Gm{Bi :1≤ i ≤ m} be the graph obtained fromGm and the trees B1,B2,...,Bm by identifying the root vertex of Bi with the ith vertex of Gm. A complete characterization is given of the eigenvalues of the Laplacian and adjacency matrices of Gm {Bi :1≤ i ≤ m} together with results about their multiplicities. Finally, these results are applied to the particular case B1 = B2 = ···= Bm.

Computing tight upper bounds on the algebraic connectivity of certain graphs

A generalized Bethe tree is a rooted unweighted tree in which vertices at the same level have the same degree. Let B be a generalized Bethe tree. The algebraic connectivity of: the generalized Bethe tree B , a tree obtained from the union of B and a tree T isomorphic to a subtree of B such that the root vertex of T is the root vertex of B , a tree obtained from the union of r generalized Bethe trees joined at their respective root vertices, a graph obtained from the cycle C r by attaching B , by its root, to each vertex of the cycle, and a tree obtained from the path P r by attaching B , by its root, to each vertex of the path, is the smallest eigenvalue of a special type of symmetric tridiagonal matrices. In this paper, we first derive a procedure to compute a tight upper bound on the smallest eigenvalue of this special type of matrices. Finally, we apply the procedure to obtain a tight upper bound on the algebraic connectivity of the above mentioned graphs.