Generalized Bethe Tree Research Articles

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.

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.

On the multiplicity of α as an eigenvalue of Aα(G) of graphs with pendant vertices

Let G be a simple undirected graph. Let 0≤α≤1. LetAα(G)=αD(G)+(1−α)A(G) where D(G) and A(G) are the diagonal matrix of the vertex degrees of G and the adjacency matrix of G, respectively. Let p(G)>0 and q(G) be the number of pendant vertices and quasi-pendant vertices of G, respectively. Let mG(α) be the multiplicity of α as an eigenvalue of Aα(G). It is proved thatmG(α)≥p(G)−q(G) with equality if each internal vertex is a quasi-pendant vertex. If there is at least one internal vertex which is not a quasi-pendant vertex, the equalitymG(α)=p(G)−q(G)+mN(α) is determined in which mN(α) is the multiplicity of α as an eigenvalue of the matrix N. This matrix is obtained from Aα(G) taking the entries corresponding to the internal vertices which are non quasi-pendant vertices. These results are applied to search for the multiplicity of α as an eigenvalue of Aα(G) when G is a path, a caterpillar, a circular caterpillar, a generalized Bethe tree or a Bethe tree. For the Bethe tree case, a simple formula for the nullity is given.

Bethe graphs attached to the vertices of a connected graph - a spectral approach

A weighted Bethe graph B is obtained from a weighted generalized Bethe tree by identifying each set of children with the vertices of a graph belonging to a family F of graphs. The operation of identifying the root vertex of each of r weighted Bethe graphs to the vertices of a connected graph of order r is introduced as the -concatenation of a family of r weighted Bethe graphs. It is shown that the Laplacian eigenvalues (when F has arbitrary graphs) as well as the signless Laplacian and adjacency eigenvalues (when the graphs in F are all regular) of the -concatenation of a family of weighted Bethe graphs can be computed (in a unified way) using the stable and low computational cost methods available for the determination of the eigenvalues of symmetric tridiagonal matrices. Unlike the previous results already obtained on this topic, the more general context of families of distinct weighted Bethe graphs is herein considered.

Eigenvalues of certain weighted graphs joined at their roots having cliques at some levels

A generalized Bethe tree is a rooted tree in which vertices at the same level have the same degree. For i=1,2,…,p, let Bi be a generalized Bethe tree of ki levels and let Δi⊆1,2,…,ki-1 such that(1) the edges of Bi connecting vertices at consecutive levels have the same weight, and(2) for j∈Δi, each set of children of Bi at the level ki-j+1 defines a clique in which the edges have weight ui,j.For i=1,2,…,p, let Gi be the graph obtained from Bi and the cliques at the levels ki-j+1 for all j∈Δi. Let G be the graph obtained from the graphs Gi1⩽i⩽p joined at their respective roots. We give a complete characterization of the eigenvalues, including their multiplicities, of the Laplacian, signless Laplacian and adjacency matrices of the graph G. Finally, we characterize the normalized Laplacian eigenvalues when G is an unweighted graph.

A class of strong limit theorems for inhomogeneous Markov chains indexed by a generalized Bethe tree on a generalized random selection system

We study strong limit theorems for a sequence of bivariate functions for an inhomogeneous Markov chain indexed by a generalized Bethe tree on a generalized random selection system by constructing a nonnegative martingale. As corollaries, we generalize results of Yang and Ye and obtain some limit theorems for frequencies of states, ordered couples of states, and the conditional expectation of a bivariate function on a Cayley tree.

Some Shannon-McMillan Approximation Theorems for Markov Chain Field on the Generalized Bethe Tree

A class of small-deviation theorems for the relative entropy densities of arbitrary random field on the generalized Bethe tree are discussed by comparing the arbitrary measure with the Markov measure on the generalized Bethe tree. As corollaries, some Shannon-Mcmillan theorems for the arbitrary random field on the generalized Bethe tree, Markov chain field on the generalized Bethe tree are obtained.

Spectra of weighted rooted graphs having prescribed subgraphs at some levels

Let B be a weighted generalized Bethe tree of k levels (k > 1) in which nj is the number of vertices at the level k j +1 (1 � jk). Let � � {1,2,...,k 1} and F= {Gj : j 2 �}, where Gj is a prescribed weighted graph on each set of children of B at the level k j+1. In this paper, the eigenvalues of a block symmetric tridiagonal matrix of order n1 +n2 +···+nk are characterized as the eigenvalues of symmetric tridiagonal matrices of order j, 1 � jk, easily constructed from the degrees of the vertices, the weights of the edges, and the eigenvalues of the matrices associated to the family of graphs F. These results are applied to characterize the eigenvalues of the Laplacian matrix, including their multiplicities, of the graph B (F) obtained from B and all the graphs in F = {Gj : j 2 �}; and also of the signless Laplacian and adjacency matrices whenever the graphs of the family F are regular.

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.

Spectral characterization of some weighted rooted graphs with cliques

The level of a vertex in a rooted graph is one more than its distance from the root vertex. A generalized Bethe tree is a rooted tree in which vertices at the same level have the same degree. We characterize completely the eigenvalues of the Laplacian, signless Laplacian and adjacency matrices of a weighted rooted graph G obtained from a weighted generalized Bethe tree of k levels and weighted cliques in which (1) the edges connecting vertices at consecutive levels have the same weight, (2) each set of children, in one or more levels, defines a weighted clique, and (3) cliques at the same level are isomorphic. These eigenvalues are the eigenvalues of symmetric tridiagonal matrices of order j × j , 1 ⩽ j ⩽ k . Moreover, we give results on the multiplicity of the eigenvalues, on the spectral radii and on the algebraic conectivity. Finally, we apply the results to the unweighted case and some particular graphs are studied.

Spectra of copies of a generalized Bethe tree attached to any graph

A generalized Bethe tree is a rooted unweighted tree in which vertices at the same level have the same degree. Let G be any connected graph. Let G { B } be the graph obtained from G by attaching a generalized Bethe tree B , by its root, to each vertex of G . We characterize completely the eigenvalues of the signless Laplacian, Laplacian and adjacency matrices of the graph G { B } including results on the eigenvalue multiplicities. Finally, for the Laplacian and signless Laplacian matrices, we recall a procedure to compute a tight upper bound on the algebraic connectivity of G { B } as well as on the smallest eigenvalue of the signless Laplacian matrix of G { B } whenever G is a non-bipartite graph.

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.

Spectra of generalized Bethe trees attached to a path

A generalized Bethe tree is a rooted tree in which vertices at the same distance from the root have the same degree. Let P m be a path of m vertices. Let { B i : 1 ⩽ i ⩽ m } be a set of generalized Bethe trees. Let P m { B i : 1 ⩽ i ⩽ m } be the tree obtained from P m and the trees B 1 , B 2 , … , B m by identifying the root vertex of B i with the i - th vertex of P m . We give a complete characterization of the eigenvalues of the Laplacian and adjacency matrices of P m { B i : 1 ⩽ i ⩽ m } . In particular, we characterize their spectral radii and the algebraic conectivity. Moreover, we derive results concerning their multiplicities. Finally, we apply the results to the case B 1 = B 2 = … = B m .

Spectra of weighted generalized Bethe trees joined at the root

A generalized Bethe tree is a rooted tree in which vertices at the same distance from the root have the same degree. Let {Bi:1⩽i⩽m} be a set of trees such that, for i=1,2,…,m,(1)Bi is a generalized Bethe tree of ki levels,(2)the vertices of Bi at the level j have degree di,ki-j+1 for j=1,2,…,ki, and(3)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,…,ki-1.Let v{Bi:1⩽i⩽m} be the tree obtained from the union of the trees Bi joined at their respective root vertices. We give a complete characterization of the eigenvalues of the Laplacian and adjacency matrices of v{Bi:1⩽i⩽m}. Moreover, we derive results concerning their multiplicities. In particular, we characterize the spectral radii, the algebraic conectivity and the second largest Laplacian eigenvalue.

Spectral property of certain class of graphs associated with generalized Bethe trees and transitive graphs

A generalized Bethe tree is a rooted tree for which the vertices in each level having equal degree. Let Bk be a generalized Bethe tree of k level, and let T r be a connected transitive graph on r vertices. Then we obtain a graph Bk?T r from r copies of Bk and T r by appending r roots to the vertices of T r respectively. In this paper, we give a simple way to characterize the eigenvalues of the adjacency matrix A(Bk ? T r) and the Laplacian matrix L(Bk?T r) of Bk?T r by means of symmetric tridiagonal matrices of order k. We also present some structure properties of the Perron vectors of A(Bk?T r) and the Fiedler vectors of L(Bk ? T r). In addition, we obtain some results on transitive graphs.

The spectra of a graph obtained from copies of a generalized Bethe tree

We generalize the concept of a Bethe tree as follows: we say that an unweighted rooted tree is a generalized Bethe tree if in each level the vertices have equal degree. If B k is a generalized Bethe tree of k levels then we characterize completely the eigenvalues of the adjacency matrix and Laplacian matrix of a graph B k ( r ) obtained from the union of r copies of B k and the cycle C r connecting the r vertex roots. Moreover, we give results on the multiplicity of the eigenvalues, on the spectral radii and on the algebraic conectivity.