quantum ergodicity for graphs related to interval maps we prove quantum ergodicity for a family of graphs that are obtained from ergodic one dimensional maps of an interval using a procedure introduced by pakonski et al j phys a 34 9303 9317 2001 as observables we take the l 2 functions on the interval the proof is based on the periodic orbit expansion of a majorant of the quantum variance specifically given a one dimensional lebesgue measure preserving map of an interval we consider an increasingly refined sequence of partitions of the interval to this sequence we associate a sequence of graphs whose directed edges correspond to elements of the partitions and on which the classical dynamics approximates the perron frobenius operator corresponding to the map we show that except possibly for subsequences of density 0 the eigenstates of the quantum graphs equidistribute in the limit of large graphs

the spectrum of unbounded operator matrices with non diagonal domain abstract systems of linear evolution equations involving non diagonal boundary conditions yield operator matrices having non diagonal domains in products of banach spaces we show how to compute the spectrum of these matrices by using virtual matrix elements and characteristic operator functions

isospectral graphs with identical nodal counts according to a recent conjecture isospectral objects have different nodal count sequences gnutzmann et al 2005 j phys a math gen 38 8921 33 we study generalized laplacians on discrete graphs and use them to construct the first non trivial counterexamples to this conjecture in addition these examples demonstrate a surprising connection between isospectral discrete and quantum graphs

the eigenvalue problem for networks of beams abstract in this paper we consider the spectral analysis of different models of networks of euler bernoulli beams we first give the characteristic equation for the spectrum secondly in some particular situations we show that the spectrum depends only on the structure of the graph thirdly we investigate the asymptotic behaviour of the eigenvalues by proving the so called weyl s formula

the berry keating operator on l 2 mathbb r _ gt rm d x and on compact quantum graphs with general self adjoint realizations the berry keating operator berry and keating 1999 siam rev 41 236 governing the schrodinger dynamics is discussed in the hilbert space and on compact quantum graphs it has been proved that the spectrum of hbk defined on is purely continuous and thus this quantization of hbk cannot yield the hypothetical hilbert polya operator possessing as eigenvalues the nontrivial zeros of the riemann zeta function a complete classification of all self adjoint extensions of hbk acting on compact quantum graphs is given together with the corresponding secular equation in form of a determinant whose zeros determine the discrete spectrum of hbk in addition an exact trace formula and the weyl asymptotics of the eigenvalue counting function are derived furthermore we introduce the squared berry keating operator which is a special case of the black scholes operator used in financial theory of option pricing again all self adjoint extensions the corresponding secular equation the trace formula and the weyl asymptotics are derived for h2bk on compact quantum graphs while the spectra of both hbk and h2bk on any compact quantum graph are discrete their weyl asymptotics demonstrate that neither hbk nor h2bk can yield as eigenvalues the nontrivial riemann zeros some simple examples are worked out in detail

exact multiplicity for degenerate two point boundary value problems with p convex nonlinearity the exact number of positive solutions of a degenerate quasilinear two point boundary value problem is investigated for the generalization of earlier results concerning the nondegenerate case with convex nonlinearity suitably defined p convex nonlinearities are considered strictly p convex c2 functions having a non negative root are classified according to the shape of the bifurcation diagram of positive solutions versus the length of the interval we have uniqueness when f 0 0 and 0 1 or 2 solutions when f 0 0 similarly to the non degenerate case provided that the number of solutions is finite however now there may also occur a continuum of solutions connected to a dead core type phenomenon the proof of our results relies on the shooting method for the characterization of the shape of the time map in contrast to the non degenerate case the shooting method does not determine directly the number of solutions owing to the lack of uniqueness of the corresponding ivp exact conditions on the uniqueness of the ivp and in the case of non uniqueness the number and types of its local solutions are given based on this all the positive solutions of the bvp can be compiled

generalized graph theory on lie g and applications in conformal field theory each magic basis of lie g supports a generalized graph theory on lie g whose axioms include the graphs of order n as a special case on so n other examples include the sine area graphs of su n and the sine area graphs of su ini the set of all magic basis generalized graph theory pairs is not known but each possible pair corresponds to a large set of new conformal field theories

existence and nonexistence of nontrivial solutions of some nonlinear degenerate elliptic equations in connection with the maximizing problem for the functional r u u lq u lp in w01 p 0 we consider the equation div u p 2 u x u q 2 u x x 1 p resp q p e has no nontrivial resp nonnegative nontrivial solution belonging to the class p u w01 p lq xi u q 2u lp p 1 i 1 2 n w01 p provided that is star shaped the crucial point of the proof of our result is to obtain an l estimate of weak solutions and to verify a certain pohozaev type inequality for weak solutions belonging to p

the structural property of a class of vector valued hyperbolic equations and applications abstract in this paper we study the structural properties of a class of vector valued hyperbolic equations with appropriate boundary conditions including the spectrum determined growth condition we prove that the equations associate with a c 0 semigroup by the structural analysis we obtained a sufficient and necessary condition for being at least an eigenvalue on the imaginary axis in particular using the asymptotic analysis technique we prove that the spectrum of the operator determined by the equations is distributed in a strip parallel to the imaginary axis and is union of finitely many separable sets furthermore we prove that the root vectors of the operator are complete and there is a sequence of root vectors that forms a riesz basis with parentheses for the hilbert state space as applications of our results we give some concrete examples in controlled complex network of euler bernoulli beams

le nombre de solutions de certains problemes semi lineaires elliptiques abstract in this paper we compute the exact number of solutions for certain semi linear elliptic problems in the first part we derive some complementary results to a theorem of ambrosetti and prodi the global behaviour of bifurcating branches is then studied for some classes of problems in the context of a concave or a convex nonlinearity in the second part we consider the branch of positive solutions lastly in the third part in dimension one for an autonomous equation we obtain a precise description of all the solutions to these problems the proofs rely on simple eigenvalue comparison arguments

an index theory for uniformly locally finite graphs abstract an index theory for uniformly locally finite ulf graphs is developed based on the adjacency operator a acting on the space of bounded sequences defined on the vertices it turns out that the characterization by upper and lower nonnegative eigenvectors is an appropriate tool to overcome the difficulties imposed by the l setting a distinctive property of the spectral radius r a in l is the identity r sup 0 x l x 0 a x x i whilethe l 2 spectral radius r 2 of the adjacency operator satisfies r 2 inf 0 x l x 0 a x x the index i as well as other order indices can serve in classifying ulf graphs and enables connections with various graph invariants e g the chromatic number can be estimated from above by 1 r moreover results on the index i in the periodic case the regular one and for graphs having only finitely many essential ramification nodes are presented

graph theory so n current algebra and the virasoro master equation we announce an isomorphism between a set of generically irrational affine virasoro constructions onso n and the unlabelled graphs of ordern on the one hand the conformal constructions are classified by the graphs while conversely a group theoretic and conformal field theoretic identification is obtained for every graph of graph theory high level expansion provides a strong argument that each construction is unitary down to some finite critical level

spectral properties and asymptotic periodicity of flows in networks we combine functional analytical and graph theoretical methods in order to study flows in networks we show that these flows can be described by a strongly continuous operator semigroup on a banach space using perron frobenius spectral theory we then prove that this semigroup behaves asymptotically periodic

control of a network of euler bernoulli beams abstract the aim is to study the boundary controllability of a system modeling the vibrations of a network of n euler bernoulli beams connected by n vibrating point masses using the classical hilbert uniqueness method the control problem is reduced to the obtention of an observability inequality the solution is then expressed in terms of fourier series so that it is also enough to show that the distance between two consecutive large eigenvalues of the spatial operator involved in this evolution problem is superior to a minimal fixed value this property called spectral gap holds as soon as the roots of a function denoted by f and giving the asymptotic behaviour of the eigenvalues are all simple for a network of n 2 different beams this assumption on the multiplicity of the roots of f denoted by a is proved to be satisfied and controllability follows for higher values of n a numerical approach allows one to prove a in many situations and no counterexample has been found but the problem of giving a general proof of controllability remains open

riesz basis approach to the stabilization of a flexible beam with a tip mass using an abstract condition of riesz basis generation of discrete operators in the hilbert spaces we show in this paper that a sequence of generalized eigenfunctions of an euler bernoulli beam equation with a tip mass under boundary linear feedback control forms a riesz basis for the state hilbert space in the meanwhile an asymptotic expression of eigenvalues and the exponential stability are readily obtained the main results of siam j control optim 36 1998 pp 1962 1986 are concluded as a special case and the additional conditions imposed there are removed

the pauli matrices in n dimensions and finest gradings of simple lie algebras of type a n 1 properties of the lie algebra gl n c are described for a basis which is a generalization of the 2 2 pauli matrices the 3 3 case is described in detail the remarkable properties of that basis are the grading of the lie algebra it offers each grading subspace is one dimensional and the matrix group it generates it is a finite group with the center of sl n c as its commutator group

dispersion for the schrodinger equation on networks in this paper we consider the schrodinger equation on a network formed by a tree with the last generation of edges formed by infinite strips we give an explicit description of the solution of the linear schrodinger equation with constant coefficients this allows us to prove dispersive estimates which in turn are useful for solving the nonlinear schrodinger equation the proof extends also to the laminar case of positive step function coefficients having a finite number of discontinuities

the exterior matrix method for sequentially coupled fourth order equations abstract this paper introduces the exterior matrix method as a means to compute asymptotically the eigenfrequencies for the vibrations of serially connected elements which are each governed by fourth order equations the timoshenko beams and a tapered euler bernoulli beams are just two examples that are solved by this method the technique involves not just lifting the transfer matrices into the higher dimensional exterior algebra space but also lifting the original fourth order equation to generally produce a sixth order exterior equation but occasionally the exterior equation is only fifth order the original fourth order equation does not have to be solvable for this technique to work since only the approximate solutions for the exterior equation are needed to compute the eigenfrequencies asymptotically

infinite dimensional algebras and a trigonometric basis for the classical lie algebras this paper explores features of the infinite dimensional algebras that have been previously introduced in particular it is shown that the classical simple lie algebras an bn cn dn may be expressed in an egalitarian basis with trigonometric structure constants the transformation to the standard cartan weyl basis and the particularly transparent n limit that this formulation allows is provided

solving nonlinear eigenvalue problems by using p version of fem abstract two nonlinear eigenvalue problems associated with strongly nonlinear partial differential equations are considered under a dirichlet boundary condition on a square domain the p version of the finite element method is applied to evaluate the first eigenvalue and the corresponding eigenfunction the numerical results are compared with the results obtained by the h version of the fem

stabilization of the wave equation on 1 d networks in this paper we study the stabilization of the wave equation on general 1 d networks for that we transfer known observability results in the context of control problems of conservative systems see r dager and e zuazua wave propagation observation and control in 1 d flexible multi structures math appl 50 springer verlag berlin 2006 into a weighted observability estimate for dissipative systems then we use an interpolation inequality similar to the one proved in p begout and f soria j differential equations 240 2007 pp 324 356 to obtain the explicit decay estimates of the energy for smooth initial data the obtained decay rate depends on the geometric and topological properties of the network we also give some examples of particular networks in which our results apply yielding different decay rates

well posedness and symmetries of strongly coupled network equations we consider a class of evolution equations taking place on the edges of a finite network and allow for feedback effects between different possibly non adjacent edges this generalizes the setting that is common in the literature where the only considered interactions take place at the boundary i e in the nodes of the network we discuss well posedness of the associated initial value problem as well as contractivity and positivity properties of its solutions finally we discuss qualitative properties that can be formulated in terms of invariance of linear subspaces of the state space i e of symmetries of the associated physical system applications to a neurobiological model as well as to a system of linear schrodinger equations on a quantum graph are discussed

eigenvalue bracketing for discrete and metric graphs abstract we develop eigenvalue estimates for the laplacians on discrete and metric graphs using various types of boundary conditions at the vertices of the metric graph via an explicit correspondence of the equilateral metric and discrete graph spectrum also in the exceptional values of the metric graph corresponding to the dirichlet spectrum we carry over these estimates from the metric graph laplacian to the discrete case we apply the results to covering graphs and present examples where the covering graph laplacians have spectral gaps

the spectrum of the hilbert space valued second derivative with general self adjoint boundary conditions abstract we consider a large class of self adjoint elliptic problems associated with the second derivative acting on a space of vector valued functions we present and survey several results that can be obtained by means of two different approaches to the study of the associated eigenvalues problems the first more general one allows to replace a secular equation which is well known in some special cases by an abstract rank condition the second one though available in general seems to apply particularly well to a specific boundary condition the sometimes dubbed anti kirchhoff condition in the literature that arises in the theory of differential operators on graphs it also permits to discuss interesting and more direct connections between the spectrum of the differential operator and some graph theoretical quantities in particular some results on the symmetry of the spectrum in either case

stabilization of bernoulli euler beams by means of a pointwise feedback force we study the energy decay of a bernoulli euler beam which is subject to a pointwise feedback force we show that both uniform and nonuniform energy decay may occur the uniform or nonuniform decay depends on the boundary conditions in the case of nonuniform decay in the energy space we give explicit polynomial decay estimates valid for regular initial data our method consists of deducing the decay estimates from observability inequalities for the associated undamped problem via sharp trace regularity results

a characteristic equation associated to an eigenvalue problem on c2 networks abstract a characteristic equation for the spectrum of an eigenvalue problem on c 2 networks is deduced and those special cases are investigated in which the spectrum depends mainly on the structure of the abstract graph belonging to the c 2 network

generalized jacobian elliptic functions and their application to bifurcation problems associated with p laplacian abstract the jacobian elliptic functions are generalized and applied to bifurcation problems associated with p laplacian the values of bifurcation parameter and the corresponding solutions are represented in terms of common parameters and a complete description of the bifurcation diagram and a closed form representation of the corresponding solutions are obtained as a by product of the representation it turns out that a kind of solution is also a solution of an eigenvalue problem of p 2 laplacian

eigenvalue estimates for the laplacian with anti kirchhoff conditions on a metric tree abstract this paper deals with the eigenvalue estimates for laplace operators with anti kirchhoff conditions on compact trees the results include lower estimates for k involving the edge lengths or diameter of when is a star and upper estimates for k involving the average edge length or the diameter of when is a general tree we get the upper eigenvalue estimates by proving a monotonicity principle for the eigenvalues with respect to the domain graph two geometric versions of the ambartsumian theorem for are also proven

tight wavelet frames on multislice graphs we present a framework for the design of wavelet transforms tailored to data defined on multislice graphs i e multiplex or dynamic graphs graphs with multiple types of interactions are ubiquitous in real life motivating the extension of wavelets to these complex domains our framework generalizes the recently proposed spectral graph wavelet transform sgwt d hammond p vandergheynst and r gribonval wavelets on graphs via spectral graph theory appl comput harmon anal vol 30 pp 129 150 mar 2011 which is designed in the spectral frequency domain of an arbitrary finite weighted graph we extend the sgwt to form a tight frame which conserves energy in the wavelet domain and define the relationship between conventional and spectral graph wavelets we then propose a design for multislice graphs that is based on the higher order singular value decomposition hosvd a powerful tool from multilinear algebra in particular the multiple adjacency matrices are stacked to form a tensor and the hosvd decomposition provides information about its third order structure analogous to that provided by matrix factorizations we obtain a set of eigennetworks and from these graph wavelets which exploit the variability across the graphs we demonstrate the feasibility of our method 1 by capturing different orientations of a gray scale image and 2 by decomposing brain signals from functional magnetic resonance imaging we show its effectiveness to identify variability across graph edges and provide meaningful decompositions

edge connectivity and the spectral gap of combinatorial and quantum graphs we derive a number of upper and lower bounds for the first nontrivial eigenvalue of laplacians on combinatorial and quantum graph in terms of the edge connectivity i e the minimal number of edges which need to be removed to make the graph disconnected on combinatorial graphs one of the bounds corresponds to a well known inequality of fiedler of which we give a new variational proof on quantum graphs the corresponding bound generalizes a recent result of band and levy all proofs are general enough to yield corresponding estimates for the p laplacian and allow us to identify the minimizers based on the betti number of the graph we also derive upper and lower bounds on all eigenvalues which are asymptotically correct i e agree with the weyl asymptotics for the eigenvalues of the quantum graph in particular the lower bounds improve the bounds of friedlander on any given graph for all but finitely many eigenvalues while the upper bounds improve recent results of ariturk our estimates are also used to derive bounds on the eigenvalues of the normalized laplacian matrix that improve known bounds of spectral graph theory

generalized graphs and unitary irrational central charge in the superconformal master equation for each magic basis of lie g it is known that the virasoro master equation on affine g contains a generalized graph theory of conformal level families in this paper it is found that the superconformal master equation on affine g so dim g similarly contains a generalized graph theory of superconformal level families for each magic basis of g the superconformal level families satisfy linear equations on the generalized graphs and the first exact unitary irrational solutions of the superconformal master equation are obtained on the sine area graphs of g su n including the simplest unitary irrational central charges c 6nx nx 8 sin2 rs n yet observed in the program

spectrum of a network of euler bernoulli beams abstract a network of n flexible beams connected by n vibrating point masses is considered the spectrum of the spatial operator involved in this evolution problem is studied if 2 is any real number outside a discrete set of values s and if is an eigenvalue then it satisfies a characteristic equation which is given the associated eigenvectors are also characterized if 2 lies in s and if the n beams are identical same mechanical properties another characteristic equation is available it is not the case for different beams no general result can be stated some numerical examples and counterexamples are given to illustrate the impossibility of such a generalization at last the asymptotic behaviour of the eigenvalues is investigated by proving the so called weyl s formula

poincare perron s method for the dirichlet problem on stratified sets abstract we extend standard poincare perron s method to the dirichlet problem on a class of multistructures this method is based on the spherical mean theorem the construction of fundamental solutions and on harnack s inequality on such domains that we first establish

some remarks on a nonhomogeneous eigenvalue problem related to generalized trigonometric functions abstract in this paper a nonhomogeneous eigenvalue problem with p laplacian is studied we first show the sturm liouville property and derive the asymptotic expansion of eigenvalues and eigenfunctions employing the derivation of the eigenfunctions we discuss the basis property related to the eigenfunctions finally we also solve two inverse problems the inverse nodal problem and ambarzumyan problem

spectral asymptotics for quantum graphs with equal edge lengths if q is smooth on the edges of a finite metric graph with all edges of length one the eigenvalues of the schrodinger operator d2 q with standard vertex conditions have generalized asymptotic expansions to all orders

eigencurves of non definite sturm liouville problems for the p laplacian abstract the eigencurves are studied for a weighted quasi linear sturm liouville type problem of the form p y p 1 r q s sgn y y p 1 on 0 1 with sturmian type boundary conditions p being the p laplacian topics include holder classes c k and asymptotics including existence of oblique asymptotes an example illustrates precision of our holder exponents when p 2

bifurcation phenomena associated to the p laplace operator on determine la structure de l ensemble des solutions u de u x p 2 u x x f u u p 2 u sur 0 1 telles que u 0 u 1 0 ou p 1 et r on demontre que les solutions avec k zeros sont uniques quand 1 2

existence and uniqueness results for a nonlinear caputo fractional boundary value problem on a star graph abstract in this paper we study a nonlinear caputo fractional boundary value problem on a star graph by means of transformation an equivalent system of fractional boundary value problem is obtained then we establish the existence and uniqueness results by fixed point theory in particular using banach s contraction principle and schaefer s fixed point theorem we also present some examples to illustrate the application of our results

spectral analysis of the schrodinger operator on binary tree shaped networks and applications abstract in this paper we analyse the spectrum of the dissipative schrodinger operator on binary tree shaped networks as applications we study the stability of the schrodinger system using a riesz basis as well as the transfer function associated to the system

classical solvability of linear parabolic equations on networks on etudie la resolubilite classique des problemes aux valeurs limites et initiales pour des equations paraboliques lineaires generales sur des reseaux c 8 finis

surgery principles for the spectral analysis of quantum graphs we present a systematic collection of spectral surgery principles for the laplacian on a compact metric graph with any of the usual vertex conditions natural dirichlet or delta type which show

nonreal eigenvalues for second order differential operators on networks with circuits it is shown that on every finite network with at least one circuit there exist second order differential operators having an infinite number of nonreal eigenvalues the presence of nonreal eigenvalues implies that these operators cannot be selfadjoint with respect to any metric these eigenvalues reveal also the existence of oscillatory solutions for the corresponding time dependent partial differential equations

on certain second order ordinary differential equations associated with sobolev poincare type inequalities on considere l equation e i 1 n u x i p 2 u x i x i u q 2 u 0 dans u x 0 sur on donne les proprietes des solutions de e pour le cas n 1 et a b mais pour p q 1 avec p q regularite existence et unicite de solutions positives

unitary dimension reduction for a class of self adjoint extensions with applications to graph like structures abstract we consider a class of self adjoint extensions using the boundary triplet technique assuming that the associated weyl function has the special form m z m z id t n z 1 with a bounded self adjoint operator t and scalar functions m n we show that there exists a class of boundary conditions such that the spectral problem for the associated self adjoint extensions in gaps of a certain reference operator admits a unitary reduction to the spectral problem for t as a motivating example we consider differential operators on equilateral metric graphs and we describe a class of boundary conditions that admit a unitary reduction to generalized discrete laplacians

eigenvalue asymptotics for the damped wave equation on metric graphs abstract we consider the linear damped wave equation on finite metric graphs and analyse its spectral properties with an emphasis on the asymptotic behaviour of eigenvalues in the case of equilateral graphs and standard coupling conditions we show that there is only a finite number of high frequency abscissas whose location is solely determined by the averages of the damping terms on each edge we further describe some of the possible behaviour when the edge lengths are no longer necessarily equal but remain commensurate

relationship between scattering matrix and spectrum of quantum graphs abstractwe investigate the equivalence between spectral characteristics of the laplace op erator on a metric graph and the associated unitary scattering operator we provethat the statistics of level spacings and moments of observations in the eigenbasescoincide in the limit that all bond lengths approach a positive constant value 1 introduction quantum graphs have attracted much attention in recent years due both to their ap plicability as physical models and their interesting mathematical properties we referthe reader to 1 and the forthcoming volume 2 for a pot pourri of new results recentreviews dedicated to quantum graphs include 3 4 in this article we focus on one feature of interest which is the use of graph models toprobe the the universality of quantum systems one of the unsolved paradoxes of quantummechanics is the observation that a great many quantum systems are remarkably similarwhen one makes statistical observations in the semi classical r egime this manifests itselfboth in the energy levels and associated energy eigenfunctions despite a great deal ofe ort this universality is poorly understood mathematically generic quantum graphsexhibit this universal behavior and represent the most likely system for which a fullmathematically rigorous proof of this universality will rst be found important steps inthis direction have been taken in 5 a quantum graph can be de ned in two di erent but related ways a complete de scription appears in the following section one may consider a self adjoint realisationof the laplace operator or a scattering matrix approach mathematically the scatteringapproach appears to be more tractable and has formed the basis of most investigations 6 7 8 9 10 moreover the spectra arising from these quantizations are subtly di erent this is not as confusing as it might seem since the statistical properties of both versionsof the spectrum are believed to coincide when averaged over a large interval it is thepurpose of this article to put a concrete mathematical foundation behind this belief the plan of the article is as follows in the next section we give precise de nitions ofthe two ways to describe quantum graphs and in section 3 describe our main results insection 4 we describe the tools used and then present in section 5 the proofs of our results

the m matrix inverse problem for the sturm liouville equation on graphs abstractwe consider an inverse spectral problemfor sturm liouville boundaryvalue problemson a graph with formally self adjoint boundary conditions at the nodes where the giveninformation is the m matrix based on the results found in s currie b a watson m matrix asymptotics for sturm liouville problems on graphs j com appl math doi 10 1016 j cam 2007 11 019 using the green s function we prove that the poles ofthe m matrix are at the eigenvalues of the associated boundary value problem and aresimple located on the real axis and that the residue at a pole is a negative semi de nitematrix with rank equal to the multiplicity of the eigenvalue we de ne the so callednorming constants and relate them to the spectral measure and the m matrix thisenables us to recover from the m matrix the boundary conditions and the potential up to a unitary equivalence for co normal boundary conditions keywords inverse problem di erential operators on graphs m function sturm liouville 2000 msc 34a55 34b45 34b20 34l05 34b27

modeling stabilization and control of serially connected beams many flexible structures consist of a large number of components coupled end to end in the form of a chain in this paper we consider the simplest type of such structures which is formed by n seri

lap number properties for p laplacian problems investigated by lyapunov methods abstract in this work we consider a one dimensional quasilinear parabolic equation and we prove that the lap number of any solution cannot increase through orbits as the time passes if the initial data is a continuous function we deal with the lap number functional as a lyapunov function and apply lap number properties to reach an understanding on the asymptotic behavior of a particular problem

concrete method for recovering the euler characteristic of quantum graphs trace formulas play a central role in the study of spectral geometry and in particular of quantum graphs the basis of our work is the result by kurasov which links the euler characteristic of metric graphs to the spectrum of their standard laplacian these ideas were shown to be applicable even in an experimental context where only a finite number of eigenvalues from a physical realization of quantum graph can be measured in the present work we analyse sufficient hypotheses which guarantee the successful recovery of we also study how to improve the efficiency of the method and in particular how to minimise the number of eigenvalues required finally we compare our findings with numerical examples surprisingly just a few dozens of eigenvalues can be enough

nodal statistics on quantum graphs it has been suggested that the distribution of the suitably normalized number of zeros of laplacian eigenfunctions contains information about the geometry of the underlying domain we study this distribution more precisely the distribution of the nodal surplus for laplacian eigenfunctions of a metric graph the existence of the distribution is established along with its symmetry one consequence of the symmetry is that the graph s first betti number can be recovered as twice the average nodal surplus of its eigenfunctions furthermore for graphs with disjoint cycles it is proven that the distribution has a universal form it is binomial over the allowed range of values of the surplus to prove the latter result we introduce the notion of a local nodal surplus and study its symmetry and dependence properties establishing that the local nodal surpluses of disjoint cycles behave like independent bernoulli variables

multiplicity of solutions of second order linear differential equations on networks abstract an upper optimal bound is obtained for the multiplicity of solutions of second order linear differential equations on networks with general boundary and node conditions this bound depends only on the abstract topological associated graph in particular the result holds for the multiplicity of eigenvalues of the corresponding operators

mathematical aspects of vacuum energy on quantum graphs we use quantum graphs as a model to study various mathematical aspects of the vacuum energy such as convergence of periodic path expansions consistency among different methods trace formulae versus method of images and the possible connection with the underlying classical dynamics in our study we derive an expansion for the vacuum energy in terms of periodic paths on the graph and prove its convergence and smooth dependence on the bond lengths of the graph for an important special case of graphs with equal bond lengths we derive a simpler explicit formula with minor changes this formula also applies to graphs with rational up to a common factor bond lengths the main results are derived using the trace formula we also discuss an alternative approach using the method of images and prove that the results are consistent this may have important consequences for other systems since the method of images unlike the trace formula includes a sum over special bounce paths we succeed in showing that in our model bounce paths do not contribute to the vacuum energy finally we discuss the proposed possible link between the magnitude of the vacuum energy and the type chaotic versus integrable of the underlying classical dynamics within a random matrix model we calculate the variance of the vacuum energy over several ensembles and find evidence that the level repulsion leads to suppression of the vacuum energy