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 cr by attaching b by its root to each vertex of the cycle and a tree obtained from the path pr 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

improved bounds for the largest eigenvalue of trees let t be a tree with vertex set v let dv denotes the degree of v v let max dv v v let u v such that du let k eu 1 where eu is the excentricity of u for j 1 2 k 2 let j maxdv distv u j we prove that 1t maxmax2 j k 2 j 1 j j 1 1 1 1 1 and 1t maxmax2 j k 2 j 1 j 1 1 1 1 where 1 t and 1 t are the largest eigenvalue of the laplacian matrix and adjacency matrix of t respectively these bounds give better results than those obtained in d stevanovic linear algebra appl 360 2003 35 42 except if 1

an explicit formula for eigenvalues of bethe trees and upper bounds on the largest eigenvalue of any tree a bethe tree bd k is a rooted unweighted of k levels in which the root vertex has degree equal to d the vertices at level j 2 j k 1 have degree equal to d 1 and the vertices at level k are the pendant vertices in this paper we first derive an explicit formula for the eigenvalues of the adjacency matrix of bd k moreover we give the corresponding multiplicities next we derive an explicit formula for the simple nonzero eigenvalues among them the largest eigenvalue of the laplacian matrix of bd k finally we obtain upper bounds on the largest eigenvalue of the adjacency matrix and of the laplacian matrix of any tree t these upper bounds are given in terms of the largest vertex degree and the radius of t and they are attained if and only if t is a bethe tree

periodicity of grover walks on generalized bethe trees abstract 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

the spectra of a graph obtained from copies of a generalized bethe tree abstract 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

a note on the integer eigenvalues of the laplacian matrix of a balanced binary tree abstract let l b k be the laplacian matrix of an unweighted balanced binary tree b k of k levels in this note we show that 1 a 1 is an eigenvalue of l b k with multiplicity l 0 k 2 4 2 k 4l 2 if k is not a multiple of 4 or 1 l 0 k 2 4 2 k 4l 2 if k is a multiple of 4 where k 2 4 is the greatest integer not exceeding k 2 4 1 b 3 is an eigenvalue of l b k if and only if k is even 2 c 5 is an eigenvalue of l b k if and only if k is a multiple of 4

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

on the spectra of some graphs like weighted rooted trees abstract let g be a weighted rooted graph of k levels such that for j 2 k 1 each vertex at level j is adjacent to one vertex at level j 1 and all edges joining a vertex at level j with a vertex at level j 1 have the same weight where the weight is a positive real number 2 if two vertices at level j are adjacent then they are adjacent to the same vertex at level j 1 and all edges joining two vertices at level j have the same weight 3 two vertices at level j have the same degree 4 there is not a vertex at level j adjacent to others two vertices at the same level we give a complete characterization of the eigenvalues of the laplacian matrix and adjacency matrix of g they are the eigenvalues of leading principal submatrices of two nonnegative symmetric tridiagonal matrices of order k k and the roots of some polynomials related with the characteristic polynomial of the referred submatrices by application of the above mentioned results we derive an upper bound on the largest eigenvalue of a graph defined by a weighted tree and a weighted triangle attached by one of its vertices to a pendant vertex of the tree

propagation algorithms for lexicographic ordering constraints finite domain constraint programming has been used with great success to tackle a wide variety of combinatorial problems in industry and academia to apply finite domain constraint programming to a problem it is modelled by a set of constraints on a set of decision variables a common modelling pattern is the use of matrices of decision variables the rows and or columns of these matrices are often symmetric leading to redundancy in a systematic search for solutions an effective method of breaking this symmetry is to constrain the assignments of the affected rows and columns to be ordered lexicographically this paper develops an incremental propagation algorithm gaclexleq that establishes generalised arc consistency on this constraint in o n operations where n is the length of the vectors furthermore this paper shows that decomposing gaclexleq into primitive constraints available in current finite domain constraint toolkits reduces the strength or increases the cost of constraint propagation also presented are extensions and modifications to the algorithm to handle strict lexicographic ordering detection of entailment and vectors of unequal length experimental results on a number of domains demonstrate the value of gaclexleq

new upper bounds on the spectral radius of trees with the given number of vertices and maximum degree abstract this paper studies the problem of estimating the spectral radius of trees with the given number of vertices and maximum degree we obtain the new upper bounds on the spectral radius of the trees and the results are the best upper bounds expressed by the number of vertices and maximum degree at present let t v e be a tree on n vertices with maximum degree where 3 n 2 denote by t the spectral radius of t we prove that 1 if n 2 then t n 1 n 2 2 2 n 3 2 and equality holds if and only if t is an almost completely full degree tree of 3 levels 2 if 2 n 2 1 then t 2 1 and equality holds if and only if t is a completely full degree tree of 3 levels 3 if n 2 1 then t 2 1 cos 2 k 1 where k log 1 2 n 1 1 1

on the tail of the waiting time in a markov modulated m g 1 queue we show that the exponential decay parameter of the waiting time in a markov modulatedm g 1 queue is no larger than that of the correspondingm g 1 queue with averaged parameters and we give a necessary and sufficient condition for equality we also explore the effect of speeding up the modulation process a key tool is a markov modulated fluid model

locating the eigenvalues of trees abstract e give an o n method that computes for any tree t and interval how many eigenvalues of t lie within the interval our method is based on sylvester s law of inertia we use our algorithm to show that the nonzero eigenvalues of a caterpillar are simple it follows that caterpillars having b back nodes where b 2 1 are not integral we also show that among the regular caterpillars c b k formed by adjoining k legs to each of b back nodes all positive roots are in the interval k 1 k 2 and c b k is not integral if b 2

upper bound on randic energy of some graphs abstract applying the cauchy schwarz inequality we obtain a sharp upper bound on the randic energy of a bipartite graph and of graphs whose adjacency matrix is partitioned into blocks with constant row sum

towards a theory of negative dependence the fkg theorem says that the positive lattice condition an easily checkable hypothesis which holds for many natural families of events implies positive association a very useful property thus there is a natural and useful theory of positively dependent events there is as yet no corresponding theory of negatively dependent events there is however a need for such a theory this paper unfortunately contains no substantial theorems its purpose is to present examples that motivate a need for such a theory give plausibility arguments for the existence of such a theory outline a few possible directions such a theory might take and state a number of specific conjectures which pertain to the examples and to a wish list of theorems

explicit construction of the eigenvectors and eigenvalues of the graph laplacian on the cayley tree abstract a generalized fourier analysis on arbitrary graphs calls for a detailed knowledge of the eigenvectors of the graph laplacian using the symmetries of the cayley tree we recursively construct the family of eigenvectors with exponentially growing eigenspaces associated with eigenvalues in the lower part of the spectrum the spectral gap decays exponentially with the tree size for large trees the eigenvalues and eigenvectors obey recursion relations which arise from the nested geometry of the tree

on the sth laplacian eigenvalue of trees of order st 1 abstract let 1 g 2 g n g 0 be the laplacian eigenvalues of a simple undirected graph g let s 2 and t 2 be integers and let t s t be the rooted tree of three levels and order st 1 such that the vertex root has degree s the vertices in level 2 have degree t and the s t 1 pendants vertices are in level 3 we prove that s t s t max s t t is a tree of order st 1 1 2 t 1 t 2 2 t 3 this result solves a conjecture due to shao et al in j y shao l zhang x y yuan on the second laplacian eigenvalue of trees of odd order linear algebra appl 419 2006 475 485

on the characteristic polynomial of a special class of graphs and spectra of balanced trees abstract let h be a simple graph with n vertices and g be a sequence of n rooted graphs g 1 g 2 g n godsil and mckay c d godsil b d mckay a new graph product and its spectrum bull austral math soc 18 1978 21 28 defined the rooted product h g of h by g by identifying the root vertex of g i with the ith vertex of h and determined the characteristic polynomial of h g in this paper we prove a general result on the determinants of some special matrices and as a corollary determine the characteristic polynomials of adjacency and laplacian matrices of h g rojo and soto o rojo r soto the spectra of the adjacency matrix and laplacian matrix for some balanced trees linear algebra appl 403 2005 97 117 computed the characteristic polynomials and the spectrum of adjacency and laplacian matrices of a class of balanced trees as an application of our results we obtain their conclusions by a simple method

the spectra of the adjacency matrix and laplacian matrix for some balanced trees abstract let t be an unweighted rooted tree of k levels such that in each level the vertices have equal degree let dk j 1 denotes the degree of the vertices in the level j we find the eigenvalues of the adjacency matrix and of the laplacian matrix of t they are the eigenvalues of principal submatrices of two nonnegative symmetric tridiagonal matrices of order k k the codiagonal entries for both matrices are d j 1 2 j k 1 and d k while the diagonal entries are zeros in the case of the adjacency matrix and dj 1 j k in the case of the laplacian matrix moreover we give some results concerning to the multiplicity of the above mentioned eigenvalues

tight bounds on the algebraic connectivity of bethe trees abstract a rooted bethe tree b d k is an unweighted rooted tree of k levels in which the vertex root has degree d the vertices in level 2 to level k 1 have degree d 1 and the vertices in level k have degree 1 pendant vertices in this paper we derive tight upper and lower bounds on the algebraic connectivity of 1 a bethe tree b d k and 2 a tree b d k 1 k 2 obtained by the union of two bethe trees b d k 1 and b d k 2 having in common the vertex root a useful tool in our study is the sherman morrison formula

the spectrum of the laplacian matrix of a balanced binary tree abstract let l b k be the laplacian matrix of an unweighted balanced binary tree b k of k levels we prove that spectrum of l b k is l b k j 1 k 1 t j s k where for 1 j k 1 t j is the j j principal submatrix of the tridiagonal k k matrix s k s k 1 2 0 0 2 3 2 0 2 0 3 2 0 0 2 2 we derive that the multiplicity of each eigenvalue of t j 1 j k 1 as an eigenvalue of l b k is at least 2 k j 1 finally for each t j using some results in electron j linear algebra 6 2000 62 we obtain lower and upper bounds for its smallest eigenvalue and an upper bound for its largest eigenvalue in particular we give upper bounds for the second largest eigenvalue and for the largest eigenvalue of l b k

spectra of weighted generalized bethe trees joined at the root abstract a generalized bethe tree is a rooted tree in which vertices at the same distance from the root have the same degree let b i 1 i m be a set of trees such that for i 1 2 m 1 b i is a generalized bethe tree of k i levels 2 the vertices of b i at the level j have degree d i k i j 1 for j 1 2 k i and 3 the edges of b i joining the vertices at the level j with the vertices at the level j 1 have weight w i k i j for j 1 2 k i 1 let v b i 1 i m be the tree obtained from the union of the trees b i joined at their respective root vertices we give a complete characterization of the eigenvalues of the laplacian and adjacency matrices of v b i 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

the law of the iterated logarithm for positively dependent random variables the levy s type maximal inequality is a key to establish the law of the iterated logarithm for associated random variables unfortunately this type inequality cannot be obtained for a generalization of association i e linear positive quadrant dependence because of their special dependence structure the purpose of this paper is to provide a different approach to obtain a law of the iterated logarithm for a sequence of linear positive quadrant dependent random variables

on the spectra of certain rooted trees abstract let t be an unweighted tree with vertex root v which is the union of two trees t 1 v 1 e 1 t 2 v 2 e 2 such that v 1 v 2 v and t 1 and t 2 have the property that the vertices in each of their levels have equal degree we characterize completely the eigenvalues of the adjacency matrix and of the laplacian matrix of t they are the eigenvalues of symmetric tridiagonal matrices whose entries are given in terms of the vertex degrees moreover we give some results about the multiplicity of the eigenvalues applications to some particular trees are developed

g majorization group induced cone orderings and reflection groups a vector y is g majorized by a vector x if y is an element of the convex hull of the orbit of x under the action of a group g it is known that if g is a finite reflection group then g majorization is equivalent to a group induced cone ordering in this paper it is established that if for a finite subgroup g of the orthogonal group g majorization is equivalent to a group induced cone ordering then g must be a reflection group

detecting anomalous activity on networks with the graph fourier scan statistic we consider the problem of deciding based on a single noisy measurement at each vertex of a given graph whether the underlying unknown signal is constant over the graph or there exists a cluster of vertices with anomalous activation this problem is relevant to several applications such as surveillance disease outbreak detection biomedical imaging environmental monitoring etc since the activations in these problems often tend to be localized to small groups of vertices in the graphs we model such activity by a class of signals that are elevated over a possibly disconnected cluster with low cut size relative to its size we analyze the corresponding generalized likelihood ratio glr statistics and relate it to the problem of finding a sparsest cut in the graph we develop a convex relaxation of the glr statistic based on spectral graph theory which we call the graph fourier scan statistic gfss in our main theoretical result we show that the performance of the gfss depends explicitly on the spectral properties of the graph to assess the optimality of the gfss we prove an information theoretic lower bound for the detection of anomalous activity on graphs because the gfss requires the specification of a tuning parameter we develop an adaptive version of the gfss using these results we are able to characterize in a very explicit form the performance of the gfss on a few notable graph topologies we demonstrate that the gfss can efficiently detect a simulated arsenic contamination in groundwater

on the aα spectra of trees abstract 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 0 1 define the matrix a g as a g d g 1 a g this paper gives several results about the a matrices of trees in particular it is shown that if t is a tree of maximal degree then the spectral radius of a t satisfies the tight inequality a t 2 1 1 which implies previous bounds of godsil lovasz and stevanovic the proof is deduced from some new results about the a matrices of bethe trees and generalized bethe trees in addition several bounds on the spectral radius of a of general graphs are proved implying tight bounds for paths and bethe trees

majorization polyhedra and statistical testing problems abstract there are important connections between majorization and convex polyhedra both weak majorization and majorization are preorders related to certain simple convex cones we investigate the facial structure of a polyhedral cone c associated with a layered directed graph a generalization of weak majorization based on c is introduced it defines a preorder of matrices an application in statistical testing theory is discussed in some detail

on the smallest eigenvalues of the line graphs of some trees abstract 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 cvetkovic and stevanovic

the spectra of some trees and bounds for the largest eigenvalue of any tree abstract let t be an unweighted tree of k levels such that in each level the vertices have equal degree let nk j 1 and dk j 1 be the number of vertices and the degree of them in the level j we find the eigenvalues of the adjacency matrix and laplacian matrix of t for the case of two vertices in level 1 nk 2 including results concerning to their multiplicity they are the eigenvalues of leading principal submatrices of nonnegative symmetric tridiagonal matrices of order k k the codiagonal entries for these matrices are d j 1 2 j k while the diagonal entries are 0 0 1 in the case of the adjacency matrix and d1 d2 dk 1 dk 1 in the case of the laplacian matrix finally we use these results to find improved upper bounds for the largest eigenvalue of the adjacency matrix and of the laplacian matrix of any given tree

order restricted statistical tests on multinomial and poisson parameters the starshaped restriction abstract likelihood ratio statistics for i testing the homogeneity of a collection of multinomial parameters against the alternative which accounts for the restriction that those parameters are starshaped cf shaked ann statist 1979 and for ii testing the null hypothesis that this parameter vector is starshaped are considered for both tests the asymptotic distribution of the test statistic under the null hypothesis is a version of the chi bar square distribution analogous tests on a collection of poisson means are also found to have asymptotic chi bar square distributions author

improved bounds for the laplacian energy of bethe trees abstract 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