existence of nonoscillatory solutions of higher order neutral delay difference equations with variable coefficients abstract in this paper we consider the following higher order neutral delay difference equations with positive and negative coefficients m x n cx n k p n x n r q n x n l 0 n n 0 where c r m 1 k 1 r l 0 are integers and p n n n 0 and q n n n 0 are sequences of nonnegative real numbers we obtain the global results with respect to c which are some sufficient conditions for the existences of nonoscillatory solutions

nonoscillatory solutions of second order nonlinear difference equations abstract in this paper we will study the nonoscillatory solution of the second order nonlinear difference equation x n 1 1 x n 1 q n x n 1 x n 0 where q n n 0 is a nonnegative sequence and 0 new sufficient conditions are established for the existence of nonoscillatory solutions of the equation which generalize and complement the main result in y g sun oscillation and nonoscillation for second order linear difference equations appl math comput 170 2005 1095 1103

maximum laplacian energy among threshold graphs we impose some restrictions on the spectra of a threshold graph allowing us to compute its laplacian energy we then show that the pineapple with clique number 1 2n3 has largest laplacian energy among all pineapples with n vertices and among all the graphs satisfying those conditions we also find large families of threshold graphs that satisfy those restrictions and exhibit pairs of such graphs with the same laplacian energy and different laplacian spectra we show by computation that the above pineapple has maximum laplacian energy among all connected threshold graphs for all n 35

note on an upper bound for sum of the laplacian eigenvalues of a graph abstract for a simple graph g with n vertices and m edges having laplacian eigenvalues 1 g 2 g n g let s k g be the sum of k largest laplacian eigenvalues of g in this note we prove that if g is a connected graph of order n 2 with m edges having clique number and vertex covering number then s k g k 1 m 1 2 with equality if k 1 and g is the graph obtained by joining n pendant vertices with one of the vertices in k our work improves a recent work of ganie et al

eigenvalues invariant factors highest weights and schubert calculus we describe recent work of klyachko totaro knutson and tao that characterizes eigenvalues of sums of hermitian matrices and decomposition of tensor products of representations of gln c we explain related applications to invariant factors of products of matrices intersections in grassmann varieties and singular values of sums and products of arbitrary matrices recent breakthroughs primarily by a klyachko b totaro a knutson and t tao with contributions by p belkale and c woodward have led to complete solutions of several old problems involving the various notions in the title our aim here is to describe this work and especially to show how these solutions are derived from it along the way we will see that these problems are also related to other areas of mathematics including geometric invariant theory symplectic geometry and combinatorics in addition we present some related applications to singular values of arbitrary matrices although many of the theorems we state here have not appeared elsewhere their proofs are mostly soft algebra based on the hard geometric or combinatorial work of others indeed this paper emphasizes concrete elementary arguments we do give some new examples and counterexamples and raise some new open questions we have attempted to point to the sources and to some of the key partial results that had been conjectured or proved before however there is a very large literature particularly for linear algebra problems about eigenvalues singular values and invariant factors we have listed only a few of these articles from whose bibliographies we hope an interested reader can trace the history we apologize to the many whose work is not cited directly we begin in the first five sections by describing each of the problems together with some of their early histories and we state as theorems the new solutions to these problems in section 6 we describe the steps toward these solutions that were carried out before the recent breakthroughs then we discuss the recent solutions and explain how these theorems follow from the work of the above mathematicians sections 7 8 9 and 10 also contain variations and generalizations of some of the theorems stated in the first five sections as well as attributions of the theorems to their authors one of our fascinations with this subject even now that we have proofs of the theorems is the challenge to understand in a deeper way why all these subjects are received by the editors in july 1999 and in revised form january 3 2000 2000 mathematics subject classification primary 15a42 22e46 14m15 secondary 05e15 13f10 14c17 15a18 47b07 the author was partly supported by nsf grant dms9970435 c 2000 american mathematical society

an example of unitary equivalence between self adjoint extensions and their parameters abstract the spectral problem for self adjoint extensions is studied using the machinery of boundary triplets for a class of symmetric operators having weyl functions of a special type we calculate explicitly the spectral projections in the form of operator valued integrals this allows one to give a constructive proof of the fact that in certain intervals the resulting self adjoint extensions are unitarily equivalent to the parameterizing boundary operator acting in a smaller space and one is able to provide an explicit form for the associated unitary transform applications to differential operators on metric graphs and to direct sums are discussed

spectral theory of semibounded sturm liouville operators with local interactions on a discrete set we study the hamiltonians hx q with type point interactions at the centers xk on the positive half line in terms of energy forms we establish analogs of some classical results on operators hq d2 dx2 q with locally integrable potentials q lloc1 r in particular we prove that the hamiltonian hx q is self adjoint if it is lower semibounded this result completes the previous results of brasche perturbation of schrodinger hamiltonians by measures selfadjointness and semiboundedness j math phys 26 621 1985 on lower semiboundedness also we prove the analog of molchanov s discreteness criteria birman s result on stability of a continuous spectrum and investigate discreteness of a negative spectrum in the recent paper kostenko a and malamud m 1 d schrodinger operators with local point interactions on a discrete set j differ equations 249 253 2010 it was shown that the spectral properties of hx hx 0 correlate with the corresponding spectral properties of a certain clas

fixed sign solutions of second order difference equations with neumann boundary conditions in this paper we study the existence of solution of the nonlinear second order difference problem with neumann boundary conditions assuming the existence of a pair of ordered lower and upper solutions and we obtain optimal existence results for the case and even for to this end we do an exhaustive study of the values of the real parameters and for which the associated green s function to the linear operator l u u 2 2 u 1 u with neumann boundary conditions has fixed sign

extremal laplacian energy of directed trees unicyclic digraphs and bicyclic digraphs abstract let a g and d g be the adjacency matrix of a digraph g with n vertices and the diagonal matrix of vertex outdegrees of g respectively then the laplacian matrix of the digraph g is l g d g a g the laplacian energy of a digraph g is defined as l e g i 1 n i 2 by using second spectral moment where 1 2 n are all the eigenvalues of l g of g in this paper by using arc shifting operation and out star shifting operation we determine the directed trees unicyclic digraphs and bicyclic digraphs which attain maximal and minimal laplacian energy among all digraphs with n vertices respectively

the laplacian energy of threshold graphs and majorization we study the laplacian energy of threshold graphs inspired by the recent results of vinagre del vecchio justo and trevisan 22 in particular we compute the degree sequences of threshold graphs that maximize or minimize the laplacian energy for a fixed number of vertices and edges the analysis involves combinatorial methods using ferrers diagrams and ideas from majorization theory some new inequalities for threshold degree sequences are obtained in this process in the review process a referee pointed out that recently and independently helmberg and trevisan 14 obtained very similar results and we discuss this connection

a lower bound for the laplacian eigenvalues of a graph proof of a conjecture by guo we show that if j is the jth largest laplacian eigenvalue and dj is the jth largest degree 1 j n of a connected graph on n vertices then j dj j 2 1 j n 1 this settles a conjecture due to guo

on existence of positive solutions and bounded oscillations for neutral difference equations abstract a criterion for the existence of a positive solution for the first order difference equation xn cxn m pnxn k 0 pn 0 is established results are also obtained for the oscillation and nonoscillation of solutions of a second order difference equation 2 xn cxn m pnxn k where the pn are either of constant sign or are oscillatory some of the results are sharp

on spectral theory for schrodinger operators with operator valued potentials abstract given a complex separable hilbert space h we consider differential expressions of the type d 2 d x 2 v x with x a or x r here v denotes a bounded operator valued potential v b h such that v is weakly measurable and the operator norm v b h is locally integrable we consider self adjoint half line l 2 realizations h in l 2 a d x h associated with assuming a to be a regular endpoint necessitating a boundary condition of the type sin u a cos u a 0 indexed by the self adjoint operator b h in addition we study self adjoint full line l 2 realizations h of in l 2 r d x h in either case we treat in detail basic spectral theory associated with h and h including weyl titchmarsh theory green s function structure eigenfunction expansions diagonalization and a version of the spectral theorem

spectral theory of schrodinger operators with infinitely many point interactions and radial positive definite functions abstract a number of results on radial positive definite functions on r n related to schoenberg s integral representation theorem are obtained they are applied to the study of spectral properties of self adjoint realizations of two and three dimensional schrodinger operators with countably many point interactions in particular we find conditions on the configuration of point interactions such that any self adjoint realization has purely absolutely continuous non negative spectrum we also apply some results on schrodinger operators to obtain new results on completely monotone functions

signless laplacian energy of a graph and energy of a line graph abstract for a simple graph g of order n size m and with signless laplacian eigenvalues q 1 q 2 q n the signless laplacian energy q e g is defined as q e g i 1 n q i d where d 2 m n is the average vertex degree of g we obtain the lower bounds for q e g in terms of first zagreb index m 1 g maximum degree d 1 second maximum degree d 2 minimum degree d n and second minimum degree d n 1 as a consequence of these bounds we obtain several bounds for the energy e l g of the line graph l g of graph g in terms of various graph parameters like m 1 g the clique number n m etc which improve some recently known bounds

commutativity and spectra of hermitian matrices abstract if two hermitian matrices commute then the eigenvalues of their sum are just the sums of the eigenvalues of the two matrices in a suitable order examples show that the converse is not true in general in this paper partial converses are obtained the technique involves a characterization of the equality cases for weyl s inequalities moreover a new proof on the commutativity of two hermitian matrices with property l and analogous results for the product of two positive definite hermitian matrices are included

coxeter energy of graphs abstract we study the concept of the coxeter energy of graphs and digraphs quivers as an analogue of gutman s adjacency energy which has applications in theoretical chemistry and is a recently widely investigated graph invariant coxeter energy ce g of a di graph g is defined to be the sum of the absolute values of all complex eigenvalues of the coxeter matrix associated with g our main inspiration for the study comes from the coxeter formalism appearing in group theory lie theory representation theory of algebras mathematical physics and other contexts we focus on the coxeter energy of trees and we prove that the path resp the maximal star has the smallest resp the greatest coxeter energy among all trees resp two large subclasses of trees with fixed number of vertices we provide several other related results as the characterization of trees with second smallest and second greatest coxeter energy bounds for coxeter energy and general facts on coxeter spectra of graphs extending known results e g for salem trees and for certain special real coxeter eigenvalues of trees additionally we discuss few other energy like quantities for the coxeter spectra of di graphs including coxeter energy normalized by the trace of the coxeter matrix and the quantities derived from variants of coulson integral formula

improved results on brouwer s conjecture for sum of the laplacian eigenvalues of a graph abstract let g be a graph with n vertices and m edges and let s k g be the sum of the k largest laplacian eigenvalues of g it was conjectured by brouwer that s k g m k 1 2 holds for 1 k n in this paper we present several families of graphs for which brouwer s conjecture holds which improve some previously known results we also establish a new upper bound on s k g for split graphs which is tight for each k 1 2 n 1 and turns out to be better than that conjectured by brouwer

on the sum of the laplacian eigenvalues of a tree given an n vertex graph g v e the laplacian spectrum of g is the set of eigenvalues of the laplacian matrix l d a where d and a denote the diagonal matrix of vertex degrees and the adjacency matrix of g respectively in this paper we study the laplacian spectrum of trees more precisely we find a new upper bound on the sum of the k largest laplacian eigenvalues of every n vertex tree where k 1 n this result is used to establish that the n vertex star has the highest laplacian energy over all n vertex trees which answers affirmatively to a question raised by radenkovic and gutman 10

degree maximal graphs are laplacian integral abstract an integer sequence d d 1 d 2 d n is graphic if there is a graph whose degree sequence is d a graph g is maximal if its degree sequence is majorized by no other graphic sequence the laplacian matrix of g is l g d g a g where d g is the diagonal matrix of vertex degrees and a g is the 0 1 adjacency matrix the article contains an explicit algorithmic construction of all maximal graphs from which it follows e g that apart from 0 the laplacian spectrum of a maximal graph is the conjugate of its degree sequence

on the sum of the laplacian eigenvalues of a graph and brouwer s conjecture abstract for a simple graph g with n vertices m edges and having laplacian eigenvalues 1 2 n 1 n 0 let s k g i 1 k i be the sum of k largest laplacian eigenvalues of g brouwer conjectured that s k g m k 1 2 for all k 1 2 n we obtain upper bounds for s k g in terms of the clique number the vertex covering number and the diameter d of a graph g we show that brouwer s conjecture holds for certain classes of graphs the laplacian energy le g of a graph g is defined as le g i 1 n i d where d 2 m n is the average degree of g we obtain an upper bound for the laplacian energy le g of a graph g in terms of the number of vertices n the number of edges m the vertex covering number and the clique number of the graph

on the spectral theory of gesztesy seba realizations of 1 d dirac operators with point interactions on a discrete set abstract we investigate spectral properties of gesztesy seba realizations d x and d x of the 1 d dirac differential expression d with point interactions on a discrete set x x n n 1 r here n n 1 and n n 1 r the gesztesy seba realizations d x and d x are the relativistic counterparts of the corresponding schrodinger operators h x and h x with and interactions respectively we define the minimal operator d x as the direct sum of the minimal dirac operators on the intervals x n 1 x n then using the regularization procedure for direct sum of boundary triplets we construct an appropriate boundary triplet for the maximal operator d x in the case d x inf x i x j i j 0 it turns out that the boundary operators b x and b x parameterizing the realizations d x and d x are jacobi matrices these matrices substantially differ from the ones appearing in spectral theory of schrodinger operators with point interactions we show that certain spectral properties of the operators d x and d x correlate with the corresponding spectral properties of the jacobi matrices b x and b x respectively using this connection we investigate spectral properties self adjointness discreteness absolutely continuous and singular spectra of gesztesy seba realizations moreover we investigate the non relativistic limit as the velocity of light c most of our results are new even in the case d x 0

approximative solutions to difference equations of neutral type asymptotic properties of solutions to difference equations of the form m x n u n x n k a n f x n b n are studied replacing the sequence u by its limit and the right side of the equation by zero we obtain an equation which we call the fundamental equation first we investigate the space of all solutions of the fundamental equation we show that any such solution is a sum of a polynomial sequence and a product of a geometric sequence and a periodic sequence next using a new version of the krasnoselski fixed point theorem and the iterated remainder operator we establish sufficient conditions under which a given solution of the fundamental equation is an approximative solution to the above equation our approach based on the iterated remainder operator allows us to control the degree of approximation in this paper we use o ns for a given nonpositive real s as a measure of approximation

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

extremal laplacian energy of threshold graphs let g be a connected threshold graph of order n with m edges and trace t in this paper we give a lower bound on laplacian energy in terms of n m and t of g from this we determine the threshold graphs with the first four minimal laplacian energies moreover we obtain the threshold graphs with the largest and the second largest laplacian energies

spectral threshold dominance brouwer s conjecture and maximality of laplacian energy abstract the laplacian energy of a graph is the sum of the distances of the eigenvalues of the laplacian matrix of the graph to the graph s average degree the maximum laplacian energy over all graphs on n nodes and m edges is conjectured to be attained for threshold graphs we prove the conjecture to hold for graphs with the property that for each k there is a threshold graph on the same number of nodes and edges whose sum of the k largest laplacian eigenvalues exceeds that of the k largest laplacian eigenvalues of the graph we call such graphs spectrally threshold dominated these graphs include split graphs and cographs and spectral threshold dominance is preserved by disjoint unions and taking complements we conjecture that all graphs are spectrally threshold dominated this conjecture turns out to be equivalent to brouwer s conjecture concerning a bound on the sum of the k largest laplacian eigenvalues

spectral enclosures for non self adjoint extensions of symmetric operators abstract the spectral properties of non self adjoint extensions a b of a symmetric operator in a hilbert space are studied with the help of ordinary and quasi boundary triples and the corresponding weyl functions these extensions are given in terms of abstract boundary conditions involving an in general non symmetric boundary operator b in the abstract part of this paper sufficient conditions for sectoriality and m sectoriality as well as sufficient conditions for a b to have a non empty resolvent set are provided in terms of the parameter b and the weyl function special attention is paid to weyl functions that decay along the negative real line or inside some sector in the complex plane and spectral enclosures for a b are proved in this situation the abstract results are applied to elliptic differential operators with local and non local robin boundary conditions on unbounded domains to schrodinger operators with potentials of complex strengths supported on unbounded hypersurfaces or infinitely many points on the real line and to quantum graphs with non self adjoint vertex couplings

oscillation and nonoscillation for second order linear differential equations abstract new oscillation and nonoscillation theorems are obtained for the second order linear differential equation u p t u 0 where p t c 0 and p t 0 conditions only about the integrals of p t on every interval 2 n t 0 2 n 1 t 0 n 1 2 for some fixed t 0 0 are used in the results

inverse spectral theory for symmetric operators with several gaps scalar type weyl functions let s be the orthogonal sum of infinitely many pairwise unitarily equivalent symmetric operators with non zero deficiency indices let j be an open subset of r if there exists a self adjoint extension s0 of s such that j is contained in the resolvent set of s0 and the associated weyl function of the pair s s0 is monotone with respect to j then for any self adjoint operator r there exists a self adjoint extension s such that the spectral parts s j and rj are unitarily equivalent it is shown that for any extension s of s the absolutely continuous spectrum of s0 is contained in that one of s moreover for a wide class of extensions the absolutely continuous parts of s and s are even unitarily equivalent

on the laplacian eigenvalues of a graph and laplacian energy abstract let g be a simple graph with n vertices m edges maximum degree average degree d 2 m n clique number having laplacian eigenvalues 1 2 n 1 n 0 for k 1 k n let s k g i 1 k i and let 1 n 1 be the number of laplacian eigenvalues greater than or equal to average degree d in this paper we obtain a lower bound for s 1 g and an upper bound for s g in terms of m and clique number of the graph as an application we obtain the stronger bounds for the laplacian energy le g i 1 n i d which improve some well known earlier bounds

new oscillation criteria for second order linear difference equations with positive and negative coefficients the oscillation of second order neutral difference equations with positive and negative coefficients of the form d 2 x n la nx n t p nx n d q nx n s 0 l 1 is investigated we obtain many new results using the comparison between both first order and second order difference equations an example is given to show the strength of the obtained results

further developments on brouwer s conjecture for the sum of laplacian eigenvalues of graphs abstract let g be a simple graph with order n and size m and having laplacian eigenvalues 1 2 n 1 n 0 and let s k g i 1 k i be the sum of k largest laplacian eigenvalues of g brouwer conjectured that s k g m k 1 2 for all k 1 2 n we obtain upper bounds for s k g in terms of the clique number the order n and integers p 0 r 1 s 1 s 2 2 associated to the structure of the graph g we discuss brouwer s conjecture for two large families of graphs the first family of graphs is obtained from a clique of size by identifying each of its vertices to a vertex of an arbitrary c cyclic graph and the second family is composed of the graphs in which the removal of the edges of the maximal complete bipartite subgraph gives a graph each of whose non trivial components is a c cyclic graph we show among these two large families of graphs the brouwer s conjecture holds for various subfamilies of graphs depending upon the value of c the order of the c cyclic graphs the clique number of the graph the order of the maximal complete bipartite subgraph and the number of the c cyclic components of the graph

comparison theorems for second order nonlinear difference equations abstract new oscillation results are obtained for the second order nonlinear difference equation r n f x n 1 g n x n 0 and its functional form r n f x n 1 g n x n 0 the role played by the argument n on the oscillation of the functional equation is explored in particular we characterize a class of sequences n which have a harmless effect on the oscillation of this type of equations some of our results generalize improve or unify known fundamental oscillation results for several particular cases of the above equations

on a conjecture for the signless laplacian eigenvalues abstract let g be a simple graph with n vertices and e g edges and q 1 g q 2 g q n g 0 be the signless laplacian eigenvalues of g let s k g i 1 k q i g where k 1 2 n f ashraf et al conjectured that s k g e g k 1 2 for k 1 2 n in this paper we give various upper bounds for s k g and prove that this conjecture is true for the following cases connected graph with sufficiently large k unicyclic graphs and bicyclic graphs for all k and tricyclic graphs when k 3 finally we discuss whether the upper bound given in this conjecture is tight or not for c cyclic graphs and propose some problems for future research

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 laplacian energy and laplacian estrada index of random multipartite graphs abstract let g be a simple graph on n vertices and m edges and 1 2 n be the eigenvalues of the laplacian matrix of g the laplacian energy of g is defined as e l g i 1 n i 2 m n and the laplacian estrada index of g is defined as l e e g i 1 n e i 2 m n in this paper we establish asymptotic lower and upper bounds to the laplacian energy and laplacian estrada index respectively for random multipartite graphs

remarks on oscillation of second order linear difference equation in this remark we shall show that the lemma 1 and parts of the proof of theorem 1 in the paper oscillation criteria for second order linear difference equations applied mathematics and computation 145 2003 591 601 are incorrect

on brouwer s conjecture for the sum of k largest laplacian eigenvalues of graphs abstract let g be a simple graph with n vertices for 1 k n denote by s k g the sum of k largest laplacian eigenvalues of g it is conjectured by brouwer that s k g e g k 1 2 where e g is the number of edges in g this conjecture has been proved to be true for k 1 k 2 k n 1 and k n for all graphs and for certain classes of graphs such as trees unicyclic graphs and bicyclic graphs for all k 1 2 n in this paper we show that if brouwer s conjecture is true for all graphs when k p 1 p n 1 2 then it is also true for all graphs when k n p 1 from which brouwer s conjecture follows automatically for all graphs when k n 2 and k n 3 we also show that for a given graph g and its complement g if s k g e g k 1 2 holds for all k then s k g e g k 1 2 holds for all k as well which implies that brouwer s conjecture holds for the complements of trees unicyclic graphs and bicyclic graphs for all k

generalized resolvents and the boundary value problems for hermitian operators with gaps a hermitian operator a with gaps j j 1 j m is studied the self adjoint extensions which put exactly kj eigenvalues into each gap j j in particular for kj 0 1 j m the extensions preserving the gaps are described in terms of boundary conditions the generalized resolvents of the extensions with the indicated properties are described also a solvability criterion and description of all the solutions of the hamburger moment problem with supports in r j 1m j j are obtained in terms of the nevanlinna matrix

on the sum of signless laplacian eigenvalues of a graph abstract for a simple graph g let e g denote the number of edges and s k g denote the sum of the k largest eigenvalues of the signless laplacian matrix of g we conjecture that for any graph g with n vertices s k g e g k 1 2 for k 1 n we prove the conjecture for k 2 for any graph and for all k for regular graphs the conjecture is an analogous to a conjecture by a e brouwer with a similar statement but for the eigenvalues of laplacian matrices of graphs

on generalized distance energy of graphs abstract for the distance matrix d g and diagonal matrix of the vertex transmissions t r g of a simple connected graph g the generalized distance matrix d g is the convex combinations of t r g and d g and is defined as d g t r g 1 d g for 0 1 if 1 2 n are the eigenvalues of d g we define the generalized distance energy of the graph g as e d g i 1 n i 2 w g n where w g is the wiener index of g this is analogous to the energies associated with the distance laplacian and distance signless laplacian matrices of g we obtain upper and lower bounds for the generalized distance energy of graphs in terms of various parameters associated with the structure of the graph g we show that for 1 2 1 the complete bipartite graph has the minimum generalized distance energy among all connected bipartite graphs and for 0 2 n 3 n 2 the star graph has the minimum generalized distance energy among all trees

corrigendum to on the sum of the laplacian eigenvalues of a graph and brouwer s conjecture linear algebra appl 501 2016 376 389 abstract in this corrigendum we provide the missing restrictions that should have been imposed on the graph g in the statements of theorem 2 2 theorem 2 5 theorem 2 6 and hence in all the theorems of section 3 and section 4

oscillation of second order perturbed nonlinear difference equations using the riccati transformation techniques we will establish some new oscillation criteria for the second order perturbed nonlinear difference equation d a n 1 dx n 1 c f n x n g n x n dx n n 1where c 0 is a quotient of odd positive integers some comparison between our theorems and those previously known results are indicated examples are interested in the text to illustrate the relevance of our results

oscillations of delay difference equations in a critical state consider the first order delay difference equation xn i 1m pi n xn ki 0 n 0 1 2 in the case where lim infn pi n pi 0 ki 0 i 1 2 a necessary and sufficient condition for the oscillation of all solutions of the above equation is established in the critical state that the corresponding limiting equation xn i 1m pixn ki 0 n 0 1 2 admits a nonoscillatory solution it is to be pointed out that there is no result in this critical state for a delay difference equation with more than one delay argument

spectral analysis of selfadjoint elliptic differential operators dirichlet to neumann maps and abstract weyl functions abstract the spectrum of a selfadjoint second order elliptic differential operator in l 2 r n is described in terms of the limiting behavior of dirichlet to neumann maps which arise in a multi dimensional glazman decomposition and correspond to an interior and an exterior boundary value problem this leads to pde analogs of renowned facts in spectral theory of odes the main results in this paper are first derived in the more abstract context of extension theory of symmetric operators and corresponding weyl functions and are applied to the pde setting afterwards

ordering trees and graphs with few cycles by algebraic connectivity abstract several approaches for ordering graphs by spectral parameters are presented in the literature we can find graph orderings either by the greatest eigenvalue spectral radius or index or by the sum of the absolute values of the eigenvalues the energy of a graph or by the second smallest eigenvalue of the laplacian matrix the algebraic connectivity among others by considering the fact that the algebraic connectivity is related to the connectivity and shape of the graphs several structural properties of graphs relative to this parameter have been studied hence a large number of papers about ordering graphs by algebraic connectivity mainly about trees and graphs with few cycles have been published this paper surveys the significant results concerning these topics trying to focus on possible points to be investigated in order to understand the difficulties to obtain partial orderings via algebraic connectivity

classification of neutral difference equations of any order with respect to the asymptotic behavior of their solutions the asymptotic behavior of the solutions to a neutral difference equation of the email protected mx n q n x t n p n x s n 0 email protected n n 0 where t n is a general retarded argument s n is a general deviated argument p n n 0 and q n n 0 are sequences of real numbers d denotes the forward difference operator dx n x n 1 x n and d j denotes the jth forward difference operator d jx n email protected d j 1x n for j 2 3 m is studied examples illustrating the results are also given

1 d schrodinger operators with local point interactions on a discrete set abstract spectral properties of 1 d schrodinger operators h x d 2 d x 2 x n x n x x n with local point interactions on a discrete set x x n n 1 are well studied when d inf n k n x n x k 0 our paper is devoted to the case d 0 we consider h x in the framework of extension theory of symmetric operators by applying the technique of boundary triplets and the corresponding weyl functions we show that the spectral properties of h x like self adjointness discreteness and lower semiboundedness correlate with the corresponding spectral properties of certain classes of jacobi matrices based on this connection we obtain necessary and sufficient conditions for the operators h x to be self adjoint lower semibounded and discrete in the case d 0 the operators with type interactions are investigated too the obtained results demonstrate that in the case d 0 as distinguished from the case d 0 the spectral properties of the operators with and type interactions are substantially different

nonoscillation and oscillation of second order half linear difference equations this paper considers the oscillation problem for the second order half linear difference equation xn 1 r 1 xn 1 qn xn r 1xn 0 where qn n 0 is a nonnegative sequence and r 0 we establish new criteria for this equation to be nonoscillatory and oscillatory respectively when r 1 our results are complete extensions of work by zhang and zhou oscillation and nonoscillation for second order linear difference equations comp math appl 39 2000 1 7 and reduce to the results in y g sun oscillation and nonoscillation for second order linear difference equations appl math comput 170 2005 1095 1103 when r 0 our results generalize and complement sun s results in y g sun oscillation and nonoscillation for half linear second order difference equations indian j pure appl math 35 2004 133 142

characterizing trees with large laplacian energy we investigate the problem of ordering trees according to their laplacian energy more precisely given a positive integer n we find a class of cardinality approximately p n whose elements are the n vertex trees with largest laplacian energy the main tool for establishing this result is a new upper bound on the sum sk t of the k largest laplacian eigenvalues of an n vertex tree t with diameter at least four where k 2 f1 ng

asymptotic behaviour of bounded solutions of a class of higher order neutral dynamic equations in this paper necessary and sufficient conditions for the asymptotic behaviour of all bounded solutions of a class of nonlinear higher order neutral dynamic equations are stated related to various ranges of the coefficient associated with the neutral part in one of which the coefficient is allowed to oscillate and not tend to zero asymptotically the new results improve and generalize the known results stated for difference equations and differential equations and answer some problems left open for the case t r for arbitrary time scales some certain examples which the results in the literature fail to deliver an answer to are given to illustrate the applicability and significance of the new results

oscillations of higher order linear difference equations abstract consider the higher order linear difference equation where m 1 is an odd integer and p n is a sequence of nonnegative real numbers we obtain several new sufficient conditions for the oscillation of all solutions of equation examples which dwell upon the importance of our results are also included

dirichlet to neumann maps abstract weyl titchmarsh m functions and a generalized index of unbounded meromorphic operator valued functions abstract we introduce a generalized index for certain meromorphic unbounded operator valued functions the class of functions is chosen such that energy parameter dependent dirichlet to neumann maps associated to uniformly elliptic partial differential operators particularly non self adjoint schrodinger operators on bounded lipschitz domains and abstract operator valued weyl titchmarsh m functions and donoghue type m functions corresponding to closed extensions of symmetric operators belong to it the principal purpose of this paper is to prove index formulas that relate the difference of the algebraic multiplicities of the discrete eigenvalues of robin realizations of non self adjoint schrodinger operators and more abstract pairs of closed operators in hilbert spaces with the generalized index of the corresponding energy dependent dirichlet to neumann maps and abstract weyl titchmarsh m functions respectively

oscillation and nonoscillation for second order linear difference equations new oscillation and nonoscillation theorems are obtained for the second order linear difference equation 2xn 1 pnxn 0 where if pn n 1 is a real sequence with pn 0

on the unitary equivalence of absolutely continuous parts of self adjoint extensions the classical weyl von neumann theorem states that for any self adjoint operator a0 in a separable hilbert space h there exists a non unique hilbert schmidt operator c c such that the perturbed operator a0 c has purely point spectrum we are interesting whether this result remains valid for non additive perturbations by considering the set exta of self adjoint extensions of a given densely defined symmetric operator a in h and some fixed a0 a0 exta we show that the ac parts a ac and a0ac of a a exta and a0 are unitarily equivalent provided that the resolvent difference ka a i 1 a0 i 1 is compact and the weyl function m of the pair a a0 admits weak boundary limits m t w limy 0m t iy for a e t r this result generalizes the classical kato rosenblum theorem moreover it demonstrates that for such pairs a a0 the weyl von neumann theorem is in general not true in the class exta

sturm liouville boundary value problems with operator potentials and unitary equivalence consider the minimal sturm liouville operator a amin generated by the differential expression a d2dt2 t in the hilbert space l2 r h where t t 0 in h we investigate the absolutely continuous parts of different self adjoint realizations of a in particular we show that dirichlet and neumann realizations ad and an are absolutely continuous and unitary equivalent to each other and to the absolutely continuous part of the krein realization moreover if inf ess t inf t then the part a acea ad of any self adjoint realization a of a is unitarily equivalent to ad in addition we prove that the absolutely continuous part a ac of any realization a is unitarily equivalent to ad provided that the resolvent difference a i 1 ad i 1 is compact the abstract results are applied to elliptic differential expressions in the half space

extrema of graph eigenvalues abstract in 1993 hong asked what are the best bounds on the k th largest eigenvalue k g of a graph g of order n this challenging question has never been tackled for any 2 k n in the present paper tight bounds are obtained for all k 2 and even tighter bounds are obtained for the k th largest singular value k g some of these bounds are based on taylor s strongly regular graphs and others on a method of kharaghani for constructing hadamard matrices the same kind of constructions are applied to other open problems like nordhaus gaddum problems of the kind how large can k g k g be these constructions are successful also in another open question how large can the ky fan norm 1 g k g be ky fan norms of graphs generalize the concept of graph energy so this question generalizes the problem for maximum energy graphs in the final section several results and problems are restated for 1 1 matrices which seem to provide a more natural ground for such research than graphs many of the results in the paper are paired with open questions and problems for further study