a discrete nodal domain theorem for trees let g be a connected graph with n vertices and let x x1 xn be a real vector a positive negative sign graph of the vector x is a maximal connected subgraph of g on vertices xi 0 xi 0 for an eigenvalue of a generalized laplacian of a tree we characterize the maximal number of sign graphs of an eigenvector we give an o n2 time algorithm to find an eigenvector with maximum number of sign graphs and we show that finding an eigenvector with minimum number of sign graphs is an np complete problem author s abstract

dirichlet p laplacian eigenvalues and cheeger constants on symmetric graphs abstract in this paper we study eigenvalues and eigenfunctions of p laplacians with dirichlet boundary condition on graphs we characterize the first eigenfunction and the maximum eigenfunction for a bipartite graph via the sign condition by the uniqueness of the first eigenfunction of p laplacian as p 1 we identify the cheeger constant of a symmetric graph with that of the quotient graph by this approach we calculate various cheeger constants of spherically symmetric graphs

minimum vertex covers and the spectrum of the normalized laplacian on trees abstract we show that in the graph spectrum of the normalized graph laplacian on trees the eigenvalue 1 and eigenvalues near 1 are strongly related to minimum vertex covers in particular for the eigenvalue 1 its multiplicity is related to the size of a minimum vertex cover and zero entries of its eigenvectors correspond to vertices in minimum vertex covers while for eigenvalues near 1 their distance to 1 can be estimated from minimum vertex covers and for the largest eigenvalue smaller than 1 the sign graphs of its eigenvectors take vertices in a minimum vertex cover as representatives

skew prime polynomial matrices the polynomial model approach abstract we examine the concept of skew primeness of polynomial matrices in terms of the associated polynomial model it is shown that skew primeness can be characterized in terms of the property of decomposition of a vector space relative to an endomorphism this basic result is then applied to the special case of nonsingular polynomial matrices we investigate the nonuniqueness of skew complements of a skew prime pair it is shown that the space of equivalence classes of skew complements is in bijective correspondence with a finite dimensional linear space finally the equivalence of the solutions to the problem of output regulation with internal stability obtained via geometric methods and via polynomial matrix techniques is shown explicitly

a lower bound for nodal count on discrete and metric graphs we study the number of nodal domains maximal connected regions on which a function has constant sign of the eigenfunctions of schrodinger operators on graphs under a certain genericity condition we show that the number of nodal domains of the nth eigenfunction is bounded below by n l where l is the number of links that distinguish the graph from a tree

stability of nodal structures in graph eigenfunctions and its relation to the nodal domain count the nodal domains of eigenvectors of the discrete schr dinger operator on simple finite and connected graphs are considered courant s well known nodal domain theorem applies in the present case and sets an upper bound to the number of nodal domains of eigenvectors arranging the spectrum as a non decreasing sequence and denoting by n the number of nodal domains of the nth eigenvector courant s theorem guarantees that the nodal deficiency n n is non negative the above applies for generic eigenvectors special care should be exercised for eigenvectors with vanishing components the main result of this work is that the nodal deficiency for generic eigenvectors is equal to a morse index of an energy functional whose value at its relevant critical points coincides with the eigenvalue the association of the nodal deficiency to the stability of an energy functional at its critical points was recently discussed in the context of quantum graphs band et al 2011 commun math phys at press doi 10 1007 s00220 011 1384 9 and dirichlet laplacian in bounded domains in berkolaiko et al 2011 arxiv 1107 3489 this work adapts this result to the discrete case the definition of the energy functional in the discrete case requires a special setting substantially different from the one used in band et al and berkolaiko et al and it is presented here in detail

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

distributed non convex first order optimization and information processing lower complexity bounds and rate optimal algorithms we consider a class of popular distributed non convex optimization problems in which agents connected by a network mathcal g collectively optimize a sum of smooth possibly non convex local objective functions we address the following question if the agents can only access the gradients of local functions what are the fastest rates that any distributed algorithms can achieve and how to achieve those rates first we show that there exist difficult problem instances such that it takes a class of distributed first order methods at least mathcal o 1 sqrt xi mathcal g times bar l epsilon communication rounds to achieve certain epsilon solution where xi mathcal g denotes the spectral gap of the graph laplacian matrix and bar l is some lipschitz constant second we propose near optimal methods whose rates match the developed lower rate bound up to a ploylog factor the key in the algorithm design is to properly embed the classical polynomial filtering techniques into modern first order algorithms to the best of our knowledge this is the first time that lower rate bounds and optimal methods have been developed for distributed non convex optimization problems

graph laplacians nodal domains and hyperplane arrangements eigenvectors of the laplacian of a graph g have received increasing attention in the recent past here we investigate their so called nodal domains i e the connected components of the maximal induced subgraphs of g on which an eigenvector psi does not change sign an analogue of courant s nodal domain theorem provides upper bounds on the number of nodal domains depending on the location of psi in the spectrum this bound however is not sharp in general in this contribution we consider the problem of computing minimal and maximal numbers of nodal domains for a particular graph the class of boolean hypercubes is discussed in detail we find that despite the simplicity of this graph class for which complete spectral information is available the computations are still non trivial nevertheless we obtained some new results and a number of conjectures author s abstract

a glimm effros dichotomy for borel equivalence relations a basic dichotomy concerning the structure of the orbit space of a transformation group has been discovered by glimm g12 in the locally compact group action case and extended by effros e 1 e2 in the polish group action case when additionally the induced equivalence relation is f it is the purpose of this paper to extend the glimm effros dichotomy to the very general context of an arbitrary borel equivalence relation not even necessarily induced by a group action despite the totally classical descriptive set theoretic nature of our result our proof requires the employment of methods of effective descriptive set theory and thus ultimately makes crucial use of computability or recursion theory on the integers

nodal domain counts and the chromatic number of graphs a function f defined on the vertices of a graph g induces nodal domains on the graph nodal domains of discrete and metric graphs are of growing interest among physicists and mathematicians in this paper several results regarding the nodal domain counts of discrete graphs are derived one such result is a global upper bound for the number of nodal domains of g in terms of its chromatic number another result is a criterion of resolution of laplacian isospectral graphs via their nodal counts several additional results are also shown

perron frobenius type results and discrete versions of nodal domain theorems abstract we prove discrete versions of nodal domain theorems in particular an eigenvector corresponding to the s th smallest eigenvalue of a graph laplacian has at most s nodal domains we compare our results to those of courant and pleijel on nodal domains of continuous laplacians and to those of fiedler on nonnegative regions of graph laplacians

zero forcing for sign patterns we introduce a new variant of zero forcing signed zero forcing the classical zero forcing number provides an upper bound on the maximum nullity of a matrix with a given graph i e zero nonzero pattern our new variant provides an analogous bound for the maximum nullity of a matrix with a given sign pattern this allows us to compute for instance the maximum nullity of a z matrix whose graph is l kn the line graph of a clique

nodal domains of eigenvectors for 1 laplacian on graphs abstract the eigenvectors for graph 1 laplacian possess some sort of localization property on one hand the characteristic function on any nodal domain of an eigenvector is again an eigenvector with the same eigenvalue on the other hand one can pack up an eigenvector for a new graph by several fundamental eigencomponents and modules with the same eigenvalue via few special techniques the courant nodal domain theorem for graphs is extended to graph 1 laplacian for strong nodal domains but for weak nodal domains it is false the notion of algebraic multiplicity is introduced in order to provide a more precise estimate of the number of independent eigenvectors a positive answer is given to a question raised in chang 2016 3 to confirm that the critical values obtained by the minimax principle may not cover all eigenvalues of graph 1 laplacian

a polynomial characterization of mathcal a mathcal b invariant and reachability subspaces based on the state space model of p fuhrmann a link is laid between the geometric approach to linear system theory as developed by w m wonham and a s morse and the approach based on polynomial matrices in particular polynomial characterizations of a b invariant and reachability subspaces are given these characterizations are used to prove the equivalence of the disturbance decoupling problem and the exact model matching problem and also to connect the polynomial matrix and the geometric approach to the construction of observers finally constructive procedures and conditions are given for computing the supremal a b invariant subspace and reachability space and for checking the solvability of the exact model matching problem

eigenvectors of the discrete laplacian on regular graphs a statistical approach in an attempt to characterize the structure of eigenvectors of random regular graphs we investigate the correlations between the components of the eigenvectors associated with different vertices in addition we provide numerical observations suggesting that the eigenvectors follow a gaussian distribution following this assumption we reconstruct some properties of the nodal structure which were observed in numerical simulations but were not explained so far dakel et al 2007 approx random pp 436 48

an eigenvector interlacing property of graphs that arise from trees by schur complementation of the laplacian abstract the literature is replete with rich connections between the structure of a graph g v e and the spectral properties of its laplacian matrix l this paper establishes similar connections between the structure of g and the laplacian l of a second graph g our interest lies in l that can be obtained from l by schur complementation in which case we say that g is partially supplied with respect to g in particular we specialize to where g is a tree with points of articulation r r and consider the partially supplied graph g derived from g by taking the schur complement with respect to r in l our results characterize how the eigenvectors of the laplacian of g relate to each other and to the structure of the tree

algebraic connectivity and degree sequences of trees we investigate the structure of trees that have minimal algebraic connectivity among all trees with a given degree sequence we show that such trees are caterpillars and that the vertex degrees are non decreasing on every path on non pendant vertices starting at the characteristic set of the fiedler vector author s abstract

graphs of given order and size and minimum algebraic connectivity the structure of connected graphs of given size and order that have minimal algebraic connectivity is investigated it is shown that they must consist of a chain of cliques moreover an upper bound for the number of maximal cliques of size 2 or larger is derived

the dual cheeger constant and spectra of infinite graphs abstract in this article we study the top of the spectrum of the normalized laplace operator on infinite graphs we introduce the dual cheeger constant and show that it controls the top of the spectrum from above and below in a similar way as the cheeger constant controls the bottom of the spectrum moreover we show that the dual cheeger constant at infinity can be used to characterize that the essential spectrum of the normalized laplace operator shrinks to one point

effective gaps are not effective quasipolynomial classical simulation of obstructed stoquastic hamiltonians all known examples suggesting an exponential separation between classical simulation algorithms and stoquastic adiabatic quantum computing stoqaqc exploit symmetries that constrain adiabatic dynamics to effective symmetric subspaces the symmetries produce large effective eigenvalue gaps which in turn make adiabatic computation efficient we present a classical algorithm to subexponentially sample from an effective subspace of any k local stoquastic hamiltonian h without a priori knowledge of its symmetries or near symmetries our algorithm maps any k local hamiltonian to a graph g v e with v o mathbf mathrm poly n mathbf where n is the number of qubits given the well known result of babai graph isomorphism in quasipolynomial time in proceedings of the forty eighth annual acm symposium on theory of computing 2016 pp 684 697 we exploit graph isomorphism to study the automorphisms of g and arrive at an algorithm quasipolynomial in v for producing samples from effective subspace eigenstates of h our results rule out exponential separations between stoqaqc and classical computation that arise from hidden symmetries in k local hamiltonians our graph representation of h is not limited to stoquastic hamiltonians and may rule out corresponding obstructions in nonstoquastic cases or be useful in studying additional properties of k local hamiltonians

a characterization of banach spaces containing it is proved that a banach space contains a subspace isomorphic to 11 if and only if it has a bounded sequence with no weak cauchy subsequence the proof yields that a sequence of subsets of a given set has a subsequence that is either convergent or boolean independent a bounded sequence of elements fn in a banach space b is said to be equivalent to the usual 11 basis provided there is a a 0 so that for all n and choices of scalars c cn

further results on polynomial characterizations of f g invariant and reachability subspaces this paper is concerned with further development of the unification between polynomial matrix and geometric theories of linear systems following the work of emre and hautus equivalence between different polynomial characterizations of f g invariant and reachability subspaces is shown explicitly several new results are given which clarify the relations between the polynomial system matrix invariant subspaces and system zeros finally a polynomial characterization of and a constructive procedure to obtain the largest stabilizability subspace in ker h are given

generalized modularity matrices various modularity matrices appeared in the recent literature on network analysis and algebraic graph theory their purpose is to allow writing as quadratic forms certain combinatorial functions appearing in the framework of graph clustering problems in this paper we put in evidence certain common traits of various modularity matrices and shed light on their spectral properties that are at the basis of various theoretical results and practical spectral type algorithms for community detection

random walks along the streets and canals in compact cities spectral analysis dynamical modularity information and statistical mechanics different models of random walks on the dual graphs of compact urban structures are considered analysis of access times between streets helps to detect the city modularity the statistical mechanics approach to the ensembles of lazy random walkers is developed the complexity of city modularity can be measured by an informationlike parameter which plays the role of an individual fingerprint of genius loci global structural properties of a city can be characterized by the thermodynamic parameters calculated in the random walk problem

nodal decompositions of graphs abstract a nodal domain of a function is a maximally connected subset of the domain for which the function does not change sign courant s nodal domain theorem gives a bound on the number of nodal domains of eigenfunctions of elliptic operators in particular the k th eigenfunction contains no more than k nodal domains we prove a generalization of courant s theorem to discrete graphs namely we show that for the k th eigenvalue of a generalized laplacian of a discrete graph there exists a set of corresponding eigenvectors such that each eigenvector can be decomposed into at most k nodal domains in addition we show this set to be of co dimension zero with respect to the entire eigenspace

dominant vertices in regulatory networks dynamics discrete time regulatory networks are dynamical systems on directed graphs with a structure that is inspired on natural systems of interacting units using a notion of determination between vertices we define sets of dominant vertices and we prove that in the asymptotic regime the projection of the dynamics on a dominant set allows us to determine the state of the whole system at all times we provide an algorithm to find sets of dominant vertices and we test its accuracy on several examples we also explore the possibility of using the dominant set characteristics as a measure of the structural complexity of networks

graph homomorphisms and nodal domains in this paper we derive some necessary spectral conditions for the existence of graph homomorphisms in which we also consider some parameters related to the corresponding eigenspaces such as nodal domains in this approach we consider the combinatorial laplacian and co laplacian as well as the adjacency matrix also we present some applications in graph decompositions where we prove a general version of fisher s inequality for g designs

a multiscale pyramid transform for graph signals multiscale transforms designed to process analog and discrete time signals and images cannot be directly applied to analyze high dimensional data residing on the vertices of a weighted graph as they do not capture the intrinsic topology of the graph data domain in this paper we adapt the laplacian pyramid transform for signals on euclidean domains so that it can be used to analyze high dimensional data residing on the vertices of a weighted graph our approach is to study existing methods and develop new methods for the four fundamental operations of graph downsampling graph reduction and filtering and interpolation of signals on graphs equipped with appropriate notions of these operations we leverage the basic multiscale constructs and intuitions from classical signal processing to generate a transform that yields both a multiresolution of graphs and an associated multiresolution of a graph signal on the underlying sequence of graphs

a study of behaviors abstract the paper presents the results of the study of behavior theory as developed by j c willems from the point of view of polynomial and rational models considering behaviors in the discrete time case to be generalizations of rational models a natural focal point becomes the concept of a behavior homomorphism we give a characterization of behavior homomorphisms and analyze their invertibility properties in terms of embeddings in unimodular polynomial matrices these results which are of intrinsic interest are then applied to the uniform derivation of a large number of results for equivalence in different classes of behavior representations to a certain extent these are generalizations of the strict system equivalence concept for the class of polynomial matrix description of systems in the style of rosenbrock a study of behavioral controllability is undertaken and gives some new insights into connections with geometric control theory

nodal domains on isospectral quantum graphs the resolution of isospectrality we present and discuss isospectral quantum graphs which are not isometric these graphs are the analogues of the isospectral domains in which were introduced recently in gordon et al 1992 bull am math soc 27 134 8 chapman 1995 am math mon 102 124 buser et al 1994 int math res not 9 391 400 okada and shudo 2001 j phys a math gen 34 5911 22 jakobson et al 2006 j comput appl math 194 141 55 and levitin et al 2006 j phys a math gen 39 2073 82 all based on sunada s construction of isospectral domains sunada t 1985 ann math 121 196 86 after presenting some of the properties of these graphs we discuss a few examples which support the conjecture that by counting the nodal domains of the corresponding eigenfunctions one can resolve the isospectral ambiguity

causal factorization and linear feedback an algebraic framework for the investigation of linear dynamic output feedback is introduced pivotal in the present theory is the problem of causal factorization i e the problem of factoring two systems over each other through a causal factor the basic issues are resolved with the aid of the new concept of latency kernels

on the spectrum of the normalized graph laplacian we investigate how the spectrum of the normalized geometric graph laplacian is affected by operations like motif dougling graph splitting or joining the multiplicity of the eigenvalue 1 or equivalently the dimension of the kernel of the adjacency matrix of the graph is of particular interest this multiplicity can be increased for instance by motif doubling

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

interlacing property of zeros of eigenvectors of schrodinger operators on trees we prove an analogue for trees of sturm s oscillation theorem on the interlacing property of zeros of eigenfunctions of a schrodinger operator let be a finite tree and a combinatorial schrodinger operator on if the eigenvectors of are ordered according to increasing eigenvalues and the vertices corresponding to zero coordinates are of degree at most two then the zeros of the linear extensions of eigenvectors have the interlacing property

disturbance decoupling problems via dynamic output feedback this paper considers the disturbance decoupling problems with or without internal stability and pole placement via dynamic output feedback using polynomial and rational matrix techniques we show that in all three problems considered the central solvability condition can be expressed as a two sided matching problem a bxc where a b and c are the polynomial system matrices of certain natural subsystems of the system model and x is to be determined over various subrings of the rational functions this matching problem can in turn be reduced to certain appropriate zero cancellation conditions on the polynomial system matrices a b and c

positive solutions of yamabe type equations with function coefficients on graphs abstract in this paper we study the yamabe type semi linear equation u a x u b x u 1 on both finite and locally finite graphs we determine certain conditions on the coefficients of the equation that guarantee existence of positive solutions in both cases some uniqueness results are also established

landscapes on spaces of trees combinatorial optimization problems defined on sets of phylogenetic trees are an important issue in computational biology for instance the problem of reconstructing a phylogeny using maximum likelihood or parsimony approaches the collection of possible phylogenetic trees is arranged as a so called robinson graph by means of the nearest neighborhood interchange move the coherent algebra and spectra of robinson graphs are discussed in some detail as their knowledge is important for an understanding of the landscape structure we consider simple model landscapes as well as landscapes arising from the maximum parsimony problem focusing on two complementary measures of ruggedness the amplitude spectrum arising from projecting the cost functions onto the eigenspaces of the underlying graph and the topology of local minima and their connecting saddle points

duality in polynomial models with some applications to geometric control theory duality is studied in the context of polynomial models for linear systems the output injection group the dual of the feedback group is studied and a polynomial characterization of c a invariant subspaces as well as of the maximal reachability subspaces contained in ker c is given

graph partitioning by eigenvectors let a be the adjacency matrix of a connected graph 9 if z is a column vector we say that a vertex of is positive nonnegative null etc if the corresponding entry of z has that property for z such that az olz we bound the number of components in the subgraph induced by positive vertices for eigenvectors z having a null element we bound the number of components in the graph induced by nonnull vertices finally bounds are established for the number of null elements in an eigenvector for the multiplicity of an eigenvalue and for the magnitudes of the second and last eigenvalues of a general or a bipartite graph

polynomial characterizations of h f invariant subspaces with applications abstract a direct study of h f invariant subspaces associated with the polynomial fractional system model z pq 1r w is undertaken the earlier results on the polynomial characterization of these subspaces are extended to cover the general case for full rank transfer matrices the smallest unobservability subspace containing im g and the largest reachability subspace in ker h are described in terms of factors of the polynomial system matrix a new polynomial model and its superspaces play the central role in all these characterizations the results obtained are applied to two measurement feedback problems in linear control theory yielding geometric interpretations for two rational matrix equations

perfect reconstruction two channel wavelet filter banks for graph structured data in this work we propose the construction of two channel wavelet filter banks for analyzing functions defined on the vertices of any arbitrary finite weighted undirected graph these graph based functions are referred to as graph signals as we build a framework in which many concepts from the classical signal processing domain such as fourier decomposition signal filtering and downsampling can be extended to graph domain especially we observe a spectral folding phenomenon in bipartite graphs which occurs during downsampling of these graphs and produces aliasing in graph signals this property of bipartite graphs allows us to design critically sampled two channel filter banks and we propose quadrature mirror filters referred to as graph qmf for bipartite graph which cancel aliasing and lead to perfect reconstruction for arbitrary graphs we present a bipartite subgraph decomposition which produces an edge disjoint collection of bipartite subgraphs graph qmfs are then constructed on each bipartite subgraph leading to multi dimensional separable wavelet filter banks on graphs our proposed filter banks are critically sampled and we state necessary and sufficient conditions for orthogonality aliasing cancellation and perfect reconstruction the filter banks are realized by chebychev polynomial approximations

autonomous subbehaviours and output nulling subspaces the aim of the present paper is to point out some connections between geometric control theory and the theory of behaviours specifically given a behaviour we analyse the set of all autonomous subbehaviours and relate those to some polynomial matrix completion problems we use this to analyse the different module structures on some vectorial polynomial spaces subsequently we apply this analysis to the study and the polynomial characterization of output nulling subspaces controlled invariant subspaces and the natural feedback induced module structures in these spaces toeplitz operators and wiener hopf factorizations play an important role

on the sign patterns of the smallest signless laplacian eigenvector abstract let h be a connected bipartite graph whose signless laplacian matrix is q h suppose that the bipartition of h is s t and that x is the eigenvector of the smallest eigenvalue of q h it is well known that x is positive and constant on s and negative and constant on t the resilience of the sign pattern of x under addition of edges into the subgraph induced by either s or t is investigated and a number of cases in which the sign pattern of x persists are described