on the optimality of the median cut spectral bisection graph partitioning method recursive spectral bisection rsb is a heuristic technique for finding a minimum cut graph bisection to use this method the second eigenvector of the laplacian of the graph is computed and from it a bisection is obtained the most common method is to use the median of the components of the second eigenvector to induce a bisection we prove here that this median cut method is optimal in the sense that the partition vector induced by it is the closest partition vector in any ls norm for s ge1 to the second eigenvector moreover we prove that the same result also holds for any m partition that is a partition into m and n m vertices when using the mth largest or smallest components of the second eigenvector

diameters and eigenvalues we derive a new upper bound for the diameter of a k regular graph g as a function of the eigenvalues of the adjacency matrix namely suppose the adjacency matrix of g has eigenvalues ai a2 a with ia i i vi21 ia i where ai k a ia21 then the diameter d g must satisfy d g we will consider families of graphs whose eigenvalues can be explicitly de termined these graphs are determined by sums or differences of vertex labels namely the pair i j being an edge depends only on the value i j or i j for directed graphs we will show that these graphs are expander graphs with small diameters by using an inequality on character sums which was recently proved by n m katz bell communications research 445 south street morristown new jersey 07960 this content downloaded from 207 46 13 111 on sat 21 may 2016 07 00 04 utc all use subject to http about jstor org terms

sparse regular random graphs spectral density and eigenvectors we examine the empirical distribution of the eigenvalues and the eigenvectors of adjacency matrices of sparse regular random graphs we find that when the degree sequence of the graph slowly increases to infinity with the number of vertices the empirical spectral distribution converges to the semicircle law moreover we prove concentration estimates on the number of eigenvalues over progressively smaller intervals we also show that with high probability all the eigenvectors are delocalized

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

reduction of attributes in averaged similarities abstract similarity relations may be constructed from a set of fuzzy attributes each fuzzy attribute generates a simple similarity and these simple similarities are combined into a complex similarity afterwards the representation theorem establishes one such way of combining similarities while averaging them is a different and more realistic approach in applied domains in this paper given an averaged similarity by a family of attributes we propose a method to find families of new attributes having fewer elements that generate the same similarity more generally the paper studies the structure of this important class of fuzzy relations

moments of the inverse participation ratio for the laplacian on finite regular graphs we investigate the rst and second moments of the inverse participation ratio ipr for all eigenvectors of the laplacian on nite random regular graphs with n vertices and degree z by exactly diagonalizing a large set of z regular graphs we nd that as n becomes large the mean of the inverse participation ratio on each graph when averaged over a large ensemble of graphs approaches the numerical value 3 this universal number is understood as the large n limit of the average of the quartic polynomial corresponding to the ipr over an appropriate n 2 dimensional hypersphere of r n the graph ensemble mean variance of the inverse participation ratio for all graph laplacian eigenvectors deviates from its continuous hypersphere average due to large graph to graph uctuations that arise from the existence of localized modes

a new paradigm for parallel adaptive meshing algorithms we present a new approach to the use of parallel computers with adaptive finite element methods this approach addresses the load balancing problem in a new way requiring far less communication than current approaches it also allows existing sequential adaptive pde codes such as pltmg and mc to run in a parallel environment without a large investment in recoding in this new approach the load balancing problem is reduced to the numerical solution of a small elliptic problem on a single processor using a sequential adaptive solver without requiring any modifications to the sequential solver the small elliptic problem is used to produce a posteriori error estimates to predict future element densities in the mesh which are then used in a weighted recursive spectral bisection of the initial mesh the bulk of the calculation then takes place independently on each processor with no communication using possibly the same sequential adaptive solver each processor adapts its region of the mesh independently and a nearly load balanced mesh distribution is usually obtained as a result of the initial weighted spectral bisection only the initial fan out of the mesh decomposition to the processors requires communication two additional steps requiring boundary exchange communication may be employed after the individual processors reach an adapted solution namely the construction of a global conforming mesh from the independent subproblems followed by a final smoothing phase using the subdomain solutions as an initial guess we present a series of convincing numerical experiments which illustrate the effectiveness of this approach the justification of the initial refinement prediction step as well as the justification of skipping the two communication intensive steps is supported by some recent j xu and a zhou math comp to appear and not so recent j a nitsche and a h schatz math comp 28 1974 pp 937 958 a h schatz and l b wahlbin math comp 31 1977 pp 414 442 a h schatz and l b wahlbin math comp 64 1995 pp 907 928 results on local a priori and a posteriori error estimation

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

graphs whose positive semi definite matrices have nullity at most two abstract let g v e be a undirected graph containing n vertices and let m g be the set of all hermitian n n matrices m m i j with m i j 0 if i and j are connected by one edge of g with m i j c if i and j are connected by at least two edges with m i j 0 if i j and i and j are not connected by an edge of g and with m i i for i 1 n a real number what is the largest nullity attained by any positive semi definite matrix m m g in this paper we characterize for t 1 and 2 those graphs g for which the maximum nullity is not greater than t

transfer potentials shape and equilibrate monetary systems abstract we analyze a monetary system of random money transfer on the basis of double entry bookkeeping without boundary conditions we do not reach a price equilibrium and violate text book formulas of economist s quantity theory mv pq to match the resulting quantity of money with the model assumption of a constant price we have to impose boundary conditions they either restrict specific transfers globally or impose transfers locally both connect through a general framework of transfer potentials we show that either restricted or imposed transfers can shape gaussian tent shape exponential boltzmann exponential pareto or periodic equilibrium distributions we derive the master equation and find its general time dependent approximate solution an equivalent of quantity theory for random money transfer under the boundary conditions of transfer potentials is given

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

expander graphs and their applications a major consideration we had in writing this survey was to make it accessible to mathematicians as well as to computer scientists since expander graphs the protagonists of our story come up in numerous and often surprising contexts in both fields but perhaps we should start with a few words about graphs in general they are of course one of the prime objects of study in discrete mathematics however graphs are among the most ubiquitous models of both natural and human made structures in the natural and social sciences they model relations among species societies companies etc in computer science they represent networks of communication data organization computational devices as well as the flow of computation and more in mathematics cayley graphs are useful in group theory graphs carry a natural metric and are therefore useful in geometry and though they are just one dimensional complexes they are useful in certain parts of topology e g knot theory in statistical physics graphs can represent local connections between interacting parts of a system as well as the dynamics of a physical process on such systems the study of these models calls then for the comprehension of the significant structural properties of the relevant graphs but are there nontrivial structural properties which are universally important expansion of a graph requires that it is simultaneously sparse and highly connected expander graphs were first defined by bassalygo and pinsker and their existence first proved by pinsker in the early 70s the property of being an expander seems significant in many of these mathematical computational and physical contexts it is not surprising that expanders are useful in the design and analysis of communication networks what is less obvious is that expanders have surprising utility in other computational settings such as in the theory of error correcting codes and the theory of pseudorandomness in mathematics we will encounter e g their role in the study of metric embeddings and in particular in work around the baum connes conjecture expansion is closely related to the convergence rates of markov chains and so they play a key role in the study of monte carlo algorithms in statistical mechanics and in a host of practical computational applications the list of such interesting and fruitful connections goes on and on with so many applications we will not even

eigenvalues and expansion of regular graphs the spectral method is the best currently known technique to prove lower bounds on expansion ramanujan graphs which have asymptotically optimal second eigenvalue are the best known explicit expanders the spectral method yielded a lower bound of k 4 on the expansion of linear sized subsets of k regular ramanujan graphs we improve the lower bound on the expansion of ramanujan graphs to approximately k 2 moreover we construct a family of k regular graphs with asymptotically optimal second eigenvalue and linear expansion equal to k 2 this shows that k 2 is the best bound one can obtain using the second eigenvalue method we also show an upper bound of roughly 1 k 1 on the average degree of linear sized induced subgraphs of ramanujan graphs this compares positively with the classical bound 2 k 1 as a byproduct we obtain improved results on random walks on expanders and construct selection networks respectively extrovert graphs of smaller size respectively degree than was previously known

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

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

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

geometric mesh partitioning implementation and experiments we investigate a method of dividing an irregular mesh into equal sized pieces with few interconnecting edges the method s novel feature is that it exploits the geometric coordinates of the mesh vertices it is based on theoretical work of miller teng thurston and vavasis who showed that certain classes of well shaped finite element meshes have good separators the geometric method is quite simple to implement we describe a sc matlab code for it in some detail the method is also quite efficient and effective we compare it with some other methods including spectral bisection

structural analysis of indistinguishability operators and related concepts given an indistinguishability operator e it is possible to obtain the set of extensional fuzzy subsets h e and the respective operators of upper and lower approximations by extensional sets f e and j e reciprocally given any of these objects the initial indistinguishability operator e can be retrieved it is well known that these concepts are in bijection in this paper we will prove that the relation underlying them is a lattice isomorphism we will also consider further operators such as natural means and finally show the robustness of the results with respect to isomorphisms of t norms

eigenvalues of graphs and a simple proof of a theorem of greenberg abstract in his ph d thesis greenberg proved that if x is the spectral radius of the universal cover x of a finite graph x then for each 0 a positive proportion depending only on x and of the eigenvalues of x have absolute value at least x in this paper we show that the same result holds true if we remove absolute from the previous result we also prove an analogue result for the smallest eigenvalues of x

state splitting for variable length graphs corresp the state splitting algorithm of adler coppersmith and hassner for graphs with edges of fixed length is extended to graphs with edges of variable lengths this has the potential to improve modulation code construction techniques although the ideas of state splitting come from dynamical systems completely graph theoretic terms are used

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

on the structure of f indistinguishability operators abstract some properties concerning the structure of f indistinguishability operators are analyzed it is shown that any of such operators is generated by a family of fuzzy subsets this result since it gives the way to construct f indistinguishabilities facilitates new applications of fuzzy relations the links between f indistinguishability operators and a kind of generalized metrics in the unit interval which are also explored are used to define the canonical generators of an f indistinguishability operator that is the fuzzy partition associated with the operator

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

the non backtracking spectrum of the universal cover of a graph a non backtracking walk on a graph h is a directed path of directed edges of h such that no edge is the inverse of its preceding edge non backtracking walks of a given length can be counted using the non backtracking adjacency matrix b indexed by h s directed edges and related to ihara s zeta function we show how to determine b s spectrum in the case where h is a tree covering a finite graph we show that when h is not regular this spectrum can have positive measure in the complex plane unlike the regular case we show that outside of b s spectrum the corresponding green function has periodic decay ratios the existence of such a ratio system can be effectively checked and is equivalent to being outside the spectrum we also prove that the spectral radius of the non backtracking walk operator on the tree covering a finite graph is exactly gr where gr is the growth rate of the tree this further motivates the definition of the graph theoretical riemann hypothesis proposed by stark and terras st finally we give experimental evidence that for a fixed finite graph h a ran dom lift of large degree has non backtracking new spectrum near that of h s universal cover this suggests a new generalization of alon s second eigenvalue conjecture

embedded in the shadow of the separator eigenvectors to the second smallest eigenvalue of the laplace matrix of a graph also known as fiedler vectors are the basic ingredient in spectral graph partitioning heuristics maximizing this second smallest eigenvalue over all nonnegative edge weightings with bounded total weight yields the absolute algebraic connectivity introduced by fiedler who proved tight connections of this value to the connectivity of the graph our objective is to gain a better understanding of the connections between separators and the eigenspace of this eigenvalue by studying the dual semidefinite optimization problem to the absolute algebraic connectivity by exploiting optimality conditions we show that this problem is equivalent to finding an embedding of the n nodes of the graph in n space so that their barycenter is the origin the distance between adjacent nodes is bounded by one and the nodes are spread as much as possible the sum of the squared norms is maximized for connected graphs we prove that for any separator in the graph at least one of the two separated node sets is embedded in the shadow with the origin being the light source of the convex hull of the separator furthermore we show that there always exists an optimal embedding whose dimension is bounded by the tree width of the graph plus one

the spectra of infinite hypertrees a model of regular infinite hypertrees is developed to mimic for hypergraphs what infinite trees do for graphs two notions of spectra or first eigenvalue are then examined for the infinite tree obtaining a precise value for the first notion and obtaining some estimates for the second the results indicate agreement of the first eigenvalue of the infinite hypertree with the second eigenvalue of a random hypergraph of the same degree to within logarithmic factors at least for the first notion of first eigenvalue author s abstract

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

interlacing eigenvalues and graphs abstract we give several old and some new applications of eigenvalue interlacing to matrices associated to graphs bounds are obtained for characteristic numbers of graphs such as the size of a maximal co clique the chromatic number the diameter and the bandwidth in terms of the eigenvalues of the standard adjacency matrix or the laplacian matrix we also deal with inequalities and regularity results concerning the structure of graphs and block designs

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

eigenfunctions of the edge based laplacian on a graph abstract in this paper we analyze the eigenfunctions of the edge based laplacian on a graph and the relationship of these functions to random walks on the graph we commence by discussing the set of eigenfunctions supported at the vertices and demonstrate the relationship of these eigenfunctions to the classical random walk on the graph then from an analysis of functions supported only on the interior of edges we develop a method for explicitly calculating the edge interior eigenfunctions of the edge based laplacian this reveals a connection between the edge based laplacian and the adjacency matrix of backtrackless random walk on the graph the edge based eigenfunctions therefore correspond to some eigenfunctions of the normalized hashimoto matrix

assets and liabilities are the momentum of particles and antiparticles displayed in feynman graphs an analogy between assets and liabilities and the momentum of particles and antiparticles called actons and passons is proposed it allows physicists to use physical methods in economy for the analysis of monetary systems and for the analysis of double entry bookkeeping economists can use it to subdivide and discuss complicated balance transactions in terms of feynman graphs which introduce the time dimension to bookkeeping within the analogy assets and liabilities come into existence by pair creation conservation of momentum is fullled whereas the conservation of energy corresponds to the regulation of a constant amount of money interest rates accelerate the particles by imposing a negative friction the statistical description of an ideal money gas is derived and the transcription to semiconductor physics is given the analogy is hoped to open a new eld for physics and to reveal new insights on monetary systems c 2001 elsevier science b v all rights reserved

spectra of lifted ramanujan graphs abstract a random n lift of a base graph g is its cover graph h on the vertices n v g where for each edge uv in g there is an independent uniform bijection and h has all edges of the form i u i v a main motivation for studying lifts is understanding ramanujan graphs and namely whether typical covers of such a graph are also ramanujan let g be a graph with largest eigenvalue 1 and let be the spectral radius of its universal cover friedman 2003 12 proved that every new eigenvalue of a random lift of g is o 1 2 1 1 2 with high probability and conjectured a bound of o 1 which would be tight by results of lubotzky and greenberg 1995 15 linial and puder 2010 17 improved friedman s bound to o 2 3 1 1 3 for d regular graphs where 1 d and 2 d 1 this translates to a bound of o d 2 3 compared to the conjectured 2 d 1 here we analyze the spectrum of a random n lift of a d regular graph whose nontrivial eigenvalues are all at most in absolute value we show that with high probability the absolute value of every nontrivial eigenvalue of the lift is o log this result is tight up to a logarithmic factor and for d 2 3 e it substantially improves the above upper bounds of friedman and of linial and puder in particular it implies that a typical n lift of a ramanujan graph is nearly ramanujan

spectral bisection of graphs and connectedness abstract we present a refinement of the work of miroslav fiedler regarding bisections of irreducible matrices we consider graph bisections as defined by the cut set consisting of characteristic edges of the eigenvector corresponding to the smallest non zero eigenvalue of the graph laplacian the so called fiedler vector we provide a simple and transparent analysis including the cases when there exist components with value zero namely we extend the class of graphs for which the fiedler vector is guaranteed to produce connected subgraphs in the bisection furthermore we show that for all connected graphs there exist fiedler vectors such that connectedness is preserved by the bisection and estimate the measure of the set of connectedness preserving fiedler vectors with respect to the set of all fiedler vectors

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

difference equations isoperimetric inequality and transience of certain random walks the difference laplacian on a square lattice in rn has been stud ied by many authors in this paper an analogous difference operator is studied for an arbitrary graph it is shown that many properties of the laplacian in the continuous setting e g the maximum principle the harnack inequality and cheeger s bound for the lowest eigenvalue hold for this difference oper ator the difference laplacian governs the random walk on a graph just as the laplace operator governs the brownian motion as an application of the theory of the difference laplacian it is shown that the random walk on a class of graphs is transient the random walks we consider are defined as follows let k be a connected graph i e a one dimensional simplicial complex for a vertex x e k let m x denote the number of edges emanating from x the probability that a particle moves from x to another vertex y e k is l m x if x and y are connected by an edge and it is zero otherwise as observed by courant friedrichs and lewy cfl for the case of a square lattice in the plane this random walk is intimately related to the difference analog of the laplacian

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

the sum over paths covariance kernel a novel covariance measure between nodes of a directed graph this work introduces a link based covariance measure between the nodes of a weighted directed graph where a cost is associated with each arc to this end a probability distribution on the usually infinite countable set of paths through the graph is defined by minimizing the total expected cost between all pairs of nodes while fixing the total relative entropy spread in the graph this results in a boltzmann distribution on the set of paths such that long high cost paths occur with a low probability while short low cost paths occur with a high probability the sum over paths sop covariance measure between nodes is then defined according to this probability distribution two nodes are considered as highly correlated if they often co occur together on the same preferably short paths the resulting covariance matrix between nodes say n nodes in total is a gram matrix and therefore defines a valid kernel on the graph it is obtained by inverting an n times n matrix depending on the costs assigned to the arcs in the same spirit a betweenness score is also defined measuring the expected number of times a node occurs on a path the proposed measures could be used for various graph mining tasks such as computing betweenness centrality semi supervised classification of nodes visualization etc as shown in section 7

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

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

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

on eigenfunctions of markov processes on trees we begin by studying the eigenvectors associated to irreducible finite birth and death processes showing that the i nontrivial eigenvector i admits a succession of i decreasing or increasing stages each of them crossing zero imbedding naturally the finite state space into a continuous segment one can unequivocally define the zeros of i which are interlaced with those of i 1 these kind of results are deduced from a general investigation of minimax multi sets dirichlet eigenproblems which leads to a direct construction of the eigenvectors associated to birth and death processes this approach can be generically extended to eigenvectors of markov processes living on trees this enables to reinterpret the eigenvalues and the eigenvectors in terms of the previous dirichlet eigenproblems and a more general conjecture is presented about related higher order cheeger inequalities finally we carefully study the geometric structure of the eigenspace associated to the spectral gap on trees

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

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

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

permanental roots and the star degree of a graph abstract for undirected graphs without loops or multiple edges we define the star degree of a graph and prove that it is equal to the multiplicity of the root 1 of per xi b where b d a considering bipartite graphs we prove that per xi b per xi l where l d a and consequently that the star degree of a bipartite graph can also be characterized by the multiplicity of the root 1 of per xi l

nontrivial bookkeeping a mechanical perspective abstract although economic practice is measured by bookkeeping economic theory neglects its role we want to bridge this gap we analyze the structure of bookkeeping from a mechanical viewpoint and compare money with momentum bookkeeping and income statements translate to the physicist vocabulary of momentum energy and force financial transactions become feynman graphs although the realization principle enforces conservation of momentum in each currency no conservation of energy is implemented by bookkeeping we discuss the transaction axiomatics of bookkeeping from this physical viewpoint

wealth condensation in a simple model of economy abstract we introduce a simple model of economy where the time evolution is described by an equation capturing both exchange between individuals and random speculative trading in such a way that the fundamental symmetry of the economy under an arbitrary change of monetary units is insured we investigate a mean field limit of this equation and show that the distribution of wealth is of the pareto power law type the pareto behaviour of the tails of this distribution appears to be robust for finite range models as shown using both a mapping to the random directed polymer problem as well as numerical simulations in this context a phase transition between an economy dominated by a few individuals and a situation where the wealth is more evenly spread out is found an interesting outcome is that the distribution of wealth tends to be very broadly distributed when exchanges are limited either in amplitude or topologically favouring exchanges and less surprisingly increasing taxes seems to be an efficient way to reduce inequalities

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

on the eigenvectors belonging to the minimum eigenvalue of an essentially nonnegative symmetric matrix with bipartite graph we prove that if b is an essentially nonnegative symmetric matrix with minimum eigenvalue m b and if the graph of b is connected and bipartite with bipartition v1 v2 then there exists an eigenvector x belonging to m b such that xi 0 for i v1 and xi 0 for i v2 we then establish two facts concerning essentially nonnegative symmetric matrices with bipartite graph and point out how the above result can be used in conjunction with these facts to give a proof not requiring the perron frobenius theory of the following theorem of constantine 2 among all essentially nonnegative symmetric n by n matrices with minimum diagonal entry at least and maximum off diagonal entry at most m the matrix with smallest minimum eigenvalue is man in where an is the adjacency matrix of the complete bipartite graph k n2 n2

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