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

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 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

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

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

on pairs of matrices generating matrix rings and their presentations abstract let m n z be the ring of n by n matrices with integral entries and n 2 this paper studies the set g n z of pairs a b m n z 2 generating m n z as a ring we use several presentations of m n z with generators x i 1 n e i 1 i and y e 11 to obtain the following consequences 1 let k 1 the following rings have presentations with 2 generators and finitely many relations a j 1 k m m j q for any m 1 m k 2 b j 1 k m n j z where n 1 n k 2 and the same n i is repeated no more than three times 2 let d be a commutative domain of sufficiently large characteristic over which every finitely generated projective module is free we use 4 relations for x and y to describe all representations of the ring m n d into m m d for m n 3 we obtain information about the asymptotic density of g n f in m n f 2 over different fields and over the integers

a new interpretation of romkf both the approximate reduced updated kalman filter rukf and reduced order model kalman filter romkp have been used in image processing for some time this correspondence shows that the romkf can be interpreted as a special case of the approximate rukf and in so doing provides a way to estimate a needed error variance input to romkf

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

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

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

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

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

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

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

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

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

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

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