eigenvalues and 1 n odd factors abstract amahashi 1 gave a sufficient and necessary condition for the existence of 1 n odd factor in this paper for the existence of 1 n odd factors we obtain some sufficient conditions in terms of eigenvalues moreover we construct some examples which show that those results are best possible

linear transformations preserving symmetric rank one matrices let 1 be the space of n x n symmetric matrices over a commutative ring r with unit and let y y r be the subset of rank 1 that is the matrices where all 2 x 2 minors are zero our purpose is to study the invertible linear transformations z on v preserving y the study of such maps falls into two parts the first of which has already been carried out we know explicitly all the invertible t that formally preserve the vanishing of 2 x 2 minors the basic maps of that kind are in fact just the ones that first come to mind t x 1uxu for invertible u and a however as i explained in s descent theory shows that these mappings for even n automatically produce certain others as well and we call that larger set the natural transformations preserving rank d 1 an explicit description of them is given in section 1 below more precisely the theorem already known is this

linear transformations on symmetric matrices that preserve commutativity it is shown that if a nonsingular linear transformation t on the space of n square real symmetric matrices preserves the commutativity where n 3 then t a qaqt q a in for all symmetric matricesa for some scalar orthogonal matrix q and linear functional q

eigenvalues and perfect matchings we give sufficient conditions for existence of a perfect matching in a graph in terms of the eigenvalues of the laplacian matrix we also show that a distance regular graph of degree k is k edge connected

generalized quadrangles with a regular point and association schemes abstract there is a new method of constructing generalized quadrangles gqs which is based on covering of nets all gqs with a regular point can be represented in this way here we first construct from a generalized quadrangle q with a regular point a four class association scheme a q called in brief geometric it is then natural to call pseudo geometric any association scheme a with the same parameters as a q we use eigenvalue techniques and the above method of construction to give a characterization of pseudo geometric association schemes which are geometric

uniqueness of strongly regular graphs having minimal p rank abstract let be a strongly regular graph with adjacency matrix a let i be the identity matrix and let p be a prime we study the p rank of the matrices a ci for integral c and want to characterize among all strongly regular graphs with a given parameter set those for which a p rank is minimal

bounds for different spreads of line and total graphs abstract in this paper we explore some results concerning the spread of the line and the total graph of a given graph a sufficient condition for the spread of a unicyclic graph with an odd girth to be at most the spread of its line graph is presented additionally we derive an upper bound for the spread of the line graph of graphs on n vertices having a vertex edge connectivity at most a positive integer k combining techniques of interlacing of eigenvalues we derive lower bounds for the laplacian and signless laplacian spread of the total graph of a connected graph moreover for a regular graph an upper and lower bound for the spread of its total graph is given

random fibonacci sequences solutions to the random fibonacci recurrence xn 1 xn xn 1 decrease increase exponentially xn exp n for sufficiently small large in the limits 0 and we expand the lyapunov exponent in powers of and 1 respectively for the classical case of 1 we obtain exact non perturbative results in particular an invariant measure associated with ricatti variable rn xn 1 xn is shown to exhibit plateaux around all rational r

graphs with least eigenvalue 2 ten years on abstract the authors monograph spectral generalizations of line graphs was published in 2004 following the successful use of star complements to complete the classification of graphs with least eigenvalue 2 guided by citations of the book we survey progress in this area over the past decade some new observations are included

singular behaviour of certain infinite products of random 2 2 matrices the authors consider an infinite product of random matrices which appears in several physical problems in particular the ising chain in a random field the random matrices depend analytically on a parameter epsilon in such a way that for 0 they all commute under certain conditions they find that the lyapunov index of this product behaves approximately as c 2 for to 0 their approach is based on the decomposition of an exact integral equation for this problem into two reduced equations they give an expression for the exponent alpha in terms of the probability distribution of the matrices and for the proportionality constant c in terms of the solutions of the reduced integral equations in cases where exact results are available agreement is obtained the physical consequences for disordered one dimensional systems are pointed out

computation of growth rates of random sequences with multi step memory we extend the generating function approach to the computation of growth rate of random fibonacci sequences with long memory functional iteration equations are obtained and its general form is conjectured and proved based on which an asymptotic representation of the growth rate is obtained the validity of both the derived and the conjectured formula are verified upon comparison with monte carlo simulation a numerical scheme of the functional iteration is designed and implemented successfully

products of random matrices for disordered systems products of random transfer matrices are applied to low dimensional disordered systems to evaluate numerically extensive quantities such as entropy and overlap probability distribution the main advantage is the possiblity to avoid numerical differentiation the method works for arbitrary disorder distributions at any temperature

linear transformations on self adjoint matrices the preservation of rank one plus scalar abstract let m be the space of self adjoint linear maps for a nondegenerate quadratic form on a finite dimensional vector space characteristic 2 dimension at least 3 let t m m be an invertible linear transformation if t preserves the set of rank one plus scalar maps then t is of the form t x rpxp x i with one exception over z 3 z any t preserving commuting pairs also has this form with no exceptions the proof of this latter result involves computing the dimension of the space of self adjoint maps commuting with any given one

spectral classes of regular random and empirical graphs abstract we define a pseudo distance between graphs based on the spectrum of the normalized laplacian since this quantity can be computed easily or at numerically estimated it is suitable for comparing in particular large graphs numerical experiments demonstrate that the spectral distance provides a practically useful measure of graph dissimilarity the asymptotic behavior of the laplacian spectrum furthermore yields a tool for classifying families of graphs in such a way that the distance of two graphs from the same family is bounded by o 1 n in terms of size n of their vertex sets

novel computation of the growth rate of generalized random fibonacci sequences a new formulation is proposed for the computation of average growth rates of generalized random fibonacci sequences based on the new formula a novel numerical scheme is designed and successfully implemented and interesting analytic asymptotic expansions are obtained for several examples

distance regularity and the spectrum of graphs abstract we deal with the question can one see from the spectrum of a graph whether it is distance regular or not up till now the answer has not been known when has precisely four distinct eigenvalues the diameter 3 case we show that in this case the answer is negative we also give positive answers in some special situations for instance if has the spectrum of a distance regular graph with diameter 3 and 1 then is distance regular our main tools are eigenvalue techniques for partitioned matrices

eigenvalue interlacing and weight parameters of graphs eigenvalue interlacing is a versatile technique for deriving results in algebraic combinatorics in particular it has been successfully used for proving a number of results about the relation between the adjacency matrix or laplacian spectrum of a graph and some of its properties for instance some characterizations of regular partitions and bounds for some parameters such as the independence and chromatic numbers the diameter the bandwidth etc have been obtained for each parameter of a graph involving the cardinality of some vertex sets we can define its corresponding weight parameter by giving some weights that is the entries of the positive eigenvector to the vertices and replacing cardinalities by square norms the key point is that such weights regularize the graph and hence allow us to define a kind of regular partition called pseudo regular intended for general graphs here we s aow how to use interlacing for proving results about some weight parameters and pseudo regular partitions of a graph for instance generalizing a well known result of lovfisz it is shown that the weight shannon capacity 6 of a connected graph f with n vertices and adjacency matrix eigenvalues 2j 2 2 satisfies

an interlacing approach for bounding the sum of laplacian eigenvalues of graphs we apply interlacing techniques for obtaining lower and upper bounds for the sums of laplacian eigenvalues of graphs mainly we generalize two theorems of grone and grone merris providing tight bounds and studying the cases of equality as a consequence some results on well known parameters of a graph such as the maximum and minimum cuts the edge connectivity the almost dominating number and the edge isoperimetric number are derived

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

reflexive bipartite regular graphs abstract a graph is called reflexive if its second largest eigenvalue does not exceed 2 in this paper we determine all reflexive bipartite regular graphs any bipartite regular graph of degree at most 2 is reflexive as well as its bipartite complement apart from them there is a finite number of resulting graphs

random fibonacci sequences and the number 1 13198824 begin abstract for the familiar fibonacci sequence defined by f 1 f 2 1 and f n f n 1 f n 2 for n 2 f n increases exponentially with n at a rate given by the golden ratio 1 sqrt 5 2 1 61803398 ldots but for a simple modification with both additions and subtractions the it random fibonacci sequences defined by t 1 t 2 1 and for n 2 t n pm t n 1 pm t n 2 where each pm sign is independent and either or with probability 1 2 it is not even obvious if abs t n should increase with n our main result is that begin equation sqrt n abs t n rightarrow 1 13198824 ldots text as n rightarrow infty end equation with probability 1 finding the number 1 13198824 ldots involves the theory of random matrix products stern brocot division of the real line a fractal like measure a computer calculation and a rounding error analysis to validate the computer calculation end abstract

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

non hermitian localization and delocalization we study localization and delocalization in a class of non hermitian hamiltonians inspired by the problem of vortex pinning in superconductors in various simplified models we are able to obtain analytic descriptions in particular of the nonperturbative emergence of a forked structure the appearance of wings in the density of states we calculate how the localization length diverges at the localization delocalization transition we map some versions of this problem onto a random walker problem in two dimensions for a certain model we find an intricate structure in its density of states

on subdesigns of symmetric designs the proof of haemers and shrikhande uses an eigenvalue technique due to haemers 6 it is not clear whether or not 1 1 c holds provided that is a design with the parameters given above we will prove this in sect 2 by counting arguments which turn out to be much more delicate than one would expect at first this also leads to another inequality for the parameters of and which in general will not coincide with 1 1 b there are also a few further reasons to study this situation again to begin with eigenvalue techniques are less elementary than counting also a result such as theorem 1 1 should have a geometric interpretation in fact the second

commutativity preserving operators on symmetric matrices abstract if is a nonsingular linear operator on n n symmetric matrices over a formally real field n 3 and if preserves commutativity then a cu 1 au f a i where u t u and c are scalar and is a linear functional this is an extension of a result of chan and lim

on the cycle expansion for the lyapunov exponent of a product of random matrices the cycle expansion of the thermodynamical zeta function for the lyapunov exponent of a product of random matrices typically converges exponentially with the maximal cycle length mainieri 1992 phys rev lett 68 1965 in this paper we show that the convergent exponents are given by the spectrum of a properly defined evolution operator which describes how a steady distribution of vector direction is established under the action of random matrices the exponential decay terms are automatically eliminated in the cycle expansion of the spectral determinant which greatly accelerates the convergence provided all matrix elements are positive numbers as a marginal case the random fibonacci series is studied in detail and it is shown that this method is helpful

signless laplacian eigenvalue problems of nordhaus gaddum type abstract let g be a graph of order n and let q 1 g q 2 g q n g denote the signless laplacian eigenvalues of g ashraf and tayfeh rezaie 2014 3 showed that q 1 g q 1 g 3 n 4 with equality holding if and only if g or g is the star k 1 n 1 in this paper we prove that q 2 g q 2 g 2 n 4 where the equality holds if and only if g or g is k 2 p 4 or c 4 also we discuss the following problem for n 6 does q 2 g q 2 g 2 n 5 always hold we provide positive answers to this problem for the graphs with disconnected complements and the bipartite graphs and determine the graphs attaining the bound moreover we show that q 2 g q 2 g n 2 and the extremal graphs are also characterized

growth and decay of random fibonacci sequences for 0 they grow exponentially by formulating the problem as a markov chain involving random matrix products and computing its invariant measure a fractal the lyapunov constant limn x n 1 n is determined numerically for a wide range of values and its dependence on is observed to be non smooth the limit is defined in the almost sure sense this generalizes recent work of viswanath who proved 1 1 131 988 24 by a simple rescaling these results also apply to the more general random recurrence x n 1 x n x n 1 for fixed and these random recurrence relations have links with many fields including ergodic theory dynamical systems heavy tailed statistics spectral theory continued fractions and condensed matter physics

distribution of eigenvalues in non hermitian anderson models we develop a theory which describes the behavior of eigenvalues of a class of one dimensional random non hermitian operators introduced recently by hatano and nelson we prove that the eigenvalues are distributed along a curve in the complex plane an equation for the curve is derived and the density of complex eigenvalues is found in terms of spectral characteristics of a reference hermitian disordered system the generic properties of the eigenvalue distribution are discussed

theorems on partitioned matrices revisited and their applications to graph spectra some old results about spectra of partitioned matrices due to goddard and schneider or haynsworth are re proved a new result is given for the spectrum of a block stochastic matrix with the property that each off diagonal block has equal entries and each diagonal block has equal diagonal entries and equal off diagonal entries the result is applied to the study of the spectra of the usual graph matrices by partitioning the vertex set of the graph according to the neighborhood equivalence relation the concept of a reduced graph matrix is introduced the question of when n 2 is the second largest signless laplacian eigenvalue of a connected graph of order n is treated a recent conjecture posed by tam fan and zhou on graphs that maximize the signless laplacian spectral radius over all not necessarily connected graphs with given numbers of vertices and edges is refuted the laplacian spectrum of a degree maximal graph is reconsidered

the vertex edge independence number vertex edge cover number and the least eigenvalue of a graph in this paper we characterize the unique graph whose least eigenvalue attains the minimum among all graphs of a fixed order and a given vertex edge independence number or vertex edge cover number and get some bounds for the vertex edge independence number vertex edge cover number of a graph in terms of the least eigenvalue of the graph

a new class of polynomials from the spectrum of a graph and its application to bound the k independence number abstract the k independence number of a graph is the maximum size of a set of vertices at pairwise distance greater than k a graph is called k partially walk regular if the number of closed walks of a given length l k rooted at a vertex v only depends on l in particular a distance regular graph is also k partially walk regular for any k in this paper we introduce a new family of polynomials obtained from the spectrum of a graph called minor polynomials these polynomials together with the interlacing technique allow us to give tight spectral bounds on the k independence number of a k partially walk regular graph with some examples and infinite families of graphs whose bounds are tight we also show that the odd graph o l with l odd has no 1 perfect code moreover we show that our bound coincide in general with the delsarte s linear programming bound and the lovasz theta number the best ones to our knowledge in fact as a byproduct it is shown that the given bounds also apply for and the shannon capacity of a graph moreover apart from the possible interest that the minor polynomials can have our approach has the advantage of yielding closed formulas and so allowing asymptotic analysis

spectral bounds for the k independence number of a graph in this paper we obtain two spectral upper bounds for the k independence number of a graph which is the maximum size of a set of vertices at pairwise distance greater than k we construct graphs that attain equality for our first bound and show that our second bound compares favorably to previous bounds on the k independence number

commutativity preserving maps we briefly describe several techniques that have been developed to solve the problem of characterizing commutativity preserving maps we prove a new result describing the general form of injective continuous commutativity preserving maps on full matrix algebras

on the eigenvalues of distance powers of circuits abstract taking the d th distance power of a graph one adds edges between all pairs of vertices of that graph whose distance is at most d it is shown that only the numbers 3 2 1 0 1 2 d can be integer eigenvalues of a circuit distance power moreover their respective multiplicities are determined and explicit constructions for corresponding eigenspace bases containing only vectors with entries 1 0 1 are given