bulk universality for generalized wigner matrices consider n n hermitian or symmetric random matrices h where the distribution of the i j matrix element is given by a probability measure ij with a subexponential decay let sigma ij 2 be the variance for the probability measure ij with the normalization property that sum i sigma 2 ij 1 for all j under essentially the only condition that c le n sigma ij 2 le c 1 for some constant c 0 we prove that in the limit n the eigenvalue spacing statistics of h in the bulk of the spectrum coincide with those of the gaussian unitary or orthogonal ensemble gue or goe we also show that for band matrices with bandwidth m the local semicircle law holds to the energy scale m 1

characterization of oriented graphs of rank 2 abstract let d be an oriented graph i e a digraph in which all arcs are not symmetric with vertex set 1 n the adjacency matrix a a i j of d is the n n matrix defined as a i j 1 if ij is an arc of d and a i j 0 if ij is not an arc of d the rank of d is defined to be the rank of its adjacency matrix in this paper oriented graphs of rank 2 are characterized

on spectrum distribution of completely controllable linear systems this paper is concerned with the placement of the spectrum of the closed loop operator a bk resulting from use of a linear feedback control law u kx in the infinite dimensional linear control system x ax bu for a class of systems in hilbert space with certain assumptions on the spectrum of the operator a a complete characterization of the achievable spectra is obtained the proofs are carried out in an operator theoretic context

correlations of eigenvectors for non hermitian random matrix models we establish a general relation between the diagonal correlator of eigenvectors and the spectral green close quote s function for non hermitian random matrix models in the large n limit we apply this result to a number of non hermitian random matrix models and show that the outcome is in good agreement with numerical results copyright ital 1999 ital the american physical society

an upper bound of the nullity of a graph in terms of order and maximum degree abstract let g be a finite undirected graph without loops and multiple edges by and n we respectively denote the nullity the maximum vertex degree and the order of g in 10 it was proved that n 2 n 1 when g is a tree this result was generalized to a bipartite graph by 25 for a reduced bipartite graph g the above inequality was improved to n 2 2 ln 2 by 27 however the problem of bounding the nullity of an arbitrary graph g in terms of n and is left open for more than ten years in this article we aim to solve such a left problem we prove that 1 n for an arbitrary graph g with order n and maximum degree and the equality holds if and only if g is the disjoint union of some copies of k

local marchenko pastur law at the hard edge of sample covariance matrices let xn be a n n matrix whose entries are independent identically distributed complex random variables with mean zero and variance 1n we study the asymptotic spectral distribution of the eigenvalues of the covariance matrix xn xn for n we prove that the empirical density of eigenvalues in an interval e e converges to the marchenko pastur law locally on the optimal scale n e logn b and in any interval up to the hard edge logn bn2 e 4 for any 0 as a consequence we show the complete delocalization of the eigenvectors

note on the energy of regular graphs abstract for a simple graph g the energy e g is defined as the sum of the absolute values of all the eigenvalues of its adjacency matrix a g let n m respectively be the number of vertices and edges of g one well known inequality is that e g 1 n 1 2 m 1 where 1 is the spectral radius if g is k regular we have e g k k n 1 n k denote e 0 k k n 1 n k balakrishnan r balakrishnan the energy of a graph linear algebra appl 387 2004 287 295 proved that for each 0 there exist infinitely many n for each of which there exists a k regular graph g of order n with k n 1 and e g e 0 and proposed an open problem that given a positive integer n 3 and 0 does there exist a k regular graph g of order n such that e g e 0 1 in this paper we show that for each 0 there exist infinitely many such n that e g e 0 1 moreover we construct another class of simpler graphs which also supports the first assertion that e g e 0

computation of transmission zeros for distributed parameter systems the transmission zeros for a class of distributed parameter systems are defined as the limit points of the spectrum as the feedback gain increases to infinity a simple criterion which characterizes the transmission zeros different from the system operators spectrum is found the case when the transmission zeros belong to the spectrum is also discussed the presented theory is clarified with an example

on the nullity of the line graph of unicyclic graph with depth one abstract a connected graph with a unique cycle is called a unicyclic graph a unicyclic graph with depth one may be thought of as being obtained from a cycle by appending n i pendent edges on each vertex v i in the cycle c t for some integer t 3 denoted by c n 1 n 2 n t in this paper we give a complete characterization on the nullity of the line graph g l c n 1 n 2 n t as follows let m i n i 0 then i g 2 if and only if m 0 and t 0 mod 4 ii g 1 if and only if m 1 and either 1 n i 0 1 for i 1 t the length of any zero chain of n 1 n 2 n t is even and t m 0 mod 4 or 2 t 0 mod 4 and one of n 1 n 3 n t 1 0 and n 2 n 4 n t 0 must hold iii g 0 otherwise

upper and lower bounds for the energy of bipartite graphs using lagrange s multiplier rule we find upper and lower bounds of the energy of a bipartite graph g in terms of the number of vertices edges and the spectral moment of fourth order moreover the upper bound is attained in a graph g if and only if g is the graph of a symmetric balanced incomplete block design bibd also we determine the graphs for which the lower bound is sharp

on the null spaces of acyclic and unicyclic singular graphs for acyclic and unicyclic graphs we determine a necessary and sufficient condition for a graph g to be singular further it is shown that this characterization can be used to construct a basis for the null space of g

proof of a conjecture on the nullity of a connected graph in terms of order and maximum degree abstract let g be a connected graph with n vertices maximum vertex degree 2 and nullity in q zhou et al 2018 22 it was conjectured that 2 n 2 1 and the upper bound is attained if and only if g is a cycle c n with n divisible by 4 or a complete bipartite graph k n 2 n 2 in the present paper this conjecture is proved

on graphs whose energy exceeds the number of vertices abstract let g be a graph on n vertices and let 1 2 n be the eigenvalues of a 0 1 adjacency matrix of g the energy of g is e i 1 n i we characterize several classes of graphs for which e n

universal eigenvector correlations in quaternionic ginibre ensembles non hermitian random matrices enjoy non trivial correlations in the statistics of their eigenvectors we study the overlap among left and right eigenvectors in ginibre ensembles with quaternion val

relationship between the rank and the matching number of a graph abstract given a simple graph g let a g be its adjacency matrix and g be its matching number the rank of g written as r g refers to the rank of a g in this paper some relations between the rank and the matching number of a graph are studied firstly it is proved that 2 d g r g 2 g n o where d g and no are respectively the dimension of cycle space and the number of odd cycles of g secondly sharp lower bounds on both r g g and r g g are determined all the corresponding extremal graphs are characterized respectively

upper bound of skew energy of an oriented graph in terms of its skew rank abstract let g be an oriented graph with skew adjacency matrix s g the skew energy e s g of g is the sum of the norms of all eigenvalues of s g and the skew rank r s g of g is the rank of s g recently tian and wong gave a lower bound of e s g in terms of its skew rank they proved that r s g e s g and they characterized the oriented graphs which satisfy the equality in this paper we aim to establish an upper bound for skew energy of an oriented graph in terms of its skew rank and maximum vertex degree it is proved that e s g r s g for an arbitrary oriented graph g with maximum degree and the upper bound is attained if and only if g is the disjoint union of r s g 2 copies of k together with some isolated vertices where the orientation is switching equivalent to the elementary orientation of k which assigns all edges the same direction from a chromatic set to another one

a characterization of graphs with rank 5 abstract the rank of a graph g is defined to be the rank of its adjacency matrix in this paper we consider the following problem what is the structure of a connected graph g with rank 5 or equivalently what is the structure of a connected n vertex graph g whose adjacency matrix has nullity n 5 in this paper we completely characterize connected graphs g whose adjacency matrix has rank 5

the multiplicity of an arbitrary eigenvalue of a graph in terms of cyclomatic number and number of pendant vertices abstract let g be a graph with adjacency matrix a g the nullity g of g is the multiplicity of zero as an eigenvalue of a g which has received a lot of attention because of its chemical importance the multiplicity of an arbitrary eigenvalue of a g is denoted as m g in 25 the authors proved that g 2 g p g for a connected graph g with equality if and only if g is a cycle of order divisible by 4 where g e g v g 1 is the cyclomatic number of g and p g is the number of pendant vertices of g in the present paper we intend to extend this result from the nullity of g to the multiplicity of an arbitrary eigenvalue of g differing from the method in 25 by mainly applying algebraic method we prove the following result for a connected graph g m g 2 g p g for an arbitrary eigenvalue of g the equality holds if and only if g is a cycle c n and 2 cos 2 k n with k 1 2 n 2 1

triple variational principles for self adjoint operator functions for a very general class of unbounded self adjoint operator function we prove upper bounds for eigenvalues which lie within arbitrary gaps of the essential spectrum these upper bounds are given by triple variations furthermore we find conditions which imply that a point is in the resolvent set for norm resolvent continuous operator functions we show that the variational inequality becomes an equality

on the nullity of bipartite graphs abstract the nullity of a graph is defined to be the multiplicity of the eigenvalue zero in the spectrum of the adjacency matrix of the graph in this paper we obtain the nullity set of bipartite graphs of order n and characterize the bipartite graphs with nullity n 4 and the regular bipartite graphs with nullity n 6

trees with maximum nullity abstract the nullity of a graph is the multiplicity of the eigenvalue zero in its spectrum among n vertex trees the star has greatest nullity equal to n 2 we generalize this by showing that among n vertex trees whose vertex degrees do not exceed a certain value d the greatest nullity is n 2 n 1 d methods for constructing such maximum nullity trees are described

the littlewood offord problem and invertibility of random matrices abstract we prove two basic conjectures on the distribution of the smallest singular value of random n n matrices with independent entries under minimal moment assumptions we show that the smallest singular value is of order n 1 2 which is optimal for gaussian matrices moreover we give a optimal estimate on the tail probability this comes as a consequence of a new and essentially sharp estimate in the littlewood offord problem for i i d random variables x k and real numbers a k determine the probability p that the sum k a k x k lies near some number v for arbitrary coefficients a k of the same order of magnitude we show that they essentially lie in an arithmetic progression of length 1 p

statistical properties of eigenvectors in non hermitian gaussian random matrix ensembles statistical properties of eigenvectors in non hermitian random matrix ensembles are discussed with an emphasis on correlations between left and right eigenvectors two approaches are described one is an exact calculation for ginibre s ensemble in which each matrix element is an independent identically distributed gaussian complex random variable the other is a simpler calculation using n 1 as an expansion parameter where n is the rank of the random matrix this is applied to girko s ensemble consequences of eigenvector correlations which may be of physical importance in applications are also discussed it is shown that eigenvalues are much more sensitive to perturbations than in the corresponding hermitian random matrix ensembles it is also shown that in problems with time evolution governed by a non hermitian random matrix transients are controlled by eigenvector correlations

characterization of graphs whose signature equals the number of odd cycles abstract let g be a simple graph with vertex set v g and edge set e g the signature s g of g is the difference between the number of positive eigenvalues and the number of negative eigenvalues of the adjacency matrix a g in 20 it was proved that c 1 g s g c 1 g where c 1 g denotes the number of odd cycles in g a problem arises naturally what graphs have signature attaining the upper bound c 1 g resp the lower bound c 1 g in this paper we focus our attention on this problem characterizing graphs g whose signature equals c 1 g resp c 1 g

the distribution of overlaps between eigenvectors of ginibre matrices we study the overlaps between eigenvectors of nonnormal matrices they quantify the stability of the spectrum and characterize the joint eigenvalues increments under dyson type dynamics well known work by chalker and mehlig calculated the expectation of these overlaps for complex ginibre matrices for the same model we extend their results by deriving the distribution of diagonal overlaps the condition numbers and their correlations we prove i convergence of condition numbers for bulk eigenvalues to an inverse gamma distribution more generally we decompose the quenched overlap i e conditioned on eigenvalues as a product of independent random variables ii asymptotic expectation of off diagonal overlaps both for microscopic or mesoscopic separation of the corresponding eigenvalues iii decorrelation of condition numbers associated to eigenvalues at mesoscopic distance at polynomial speed in the dimension iv second moment asymptotics to identify the fluctuations order for off diagonal overlaps when the related eigenvalues are separated by any mesoscopic scale v a new formula for the correlation between overlaps for eigenvalues at microscopic distance both diagonal and off diagonal these results imply estimates on the extreme condition numbers the volume of the pseudospectrum and the diffusive evolution of eigenvalues under dyson type dynamics at equilibrium

a sharp upper bound of the nullity of a connected graph in terms of order and maximum degree abstract the nullity g of g is the multiplicity of 0 as an eigenvalue of a g in this paper we completely solve the following conjecture proposed by zhou wong and sun in linear algebra and its applications 555 2018 314 320 let g be a connected graph of order n with nullity g and the maximum degree 2 then g 2 n 2 1 the equality holds if and only if g c n n 0 m o d 4 or g k

invariant zeros of siso infinite dimensional systems the zeros of a finite dimensional system can be characterised in terms of the eigenvalues of an operator on the largest closed feedback invariant subspace this characterisation is also valid for infinite dimensional systems provided that a largest closed feedback invariant subspace exists we generalise this characterisation of the zeros to the case when the largest closed feedback invariant subspace does not exist we give an example which shows that the choice of domain of the operator on this invariant subspace is crucial to this characterisation

complex unit gain graphs of rank 2 abstract let t be the group of all complex numbers z with z 1 a complex unit gain graph or simply a t gain graph is a triple g t consisting of a graph g v e the circle group t and a gain function e t such that v i v j v j v i 1 v j v i for any adjacent vertices v i and v j the adjacency matrix a of the t gain graph g t of order n is an n n complex matrix a i j where a i j a j i v i v j if v i is adjacent to v j and a i j 0 if otherwise the rank of a t gain graph denoted by r is the rank of the adjacency matrix of yu et al 24 obtained some properties of inertia indexes of a t gain graph and they characterized the t gain unicyclic graphs with small positive or negative index lu 8 characterized the t gain bicyclic graphs with rank 2 3 or 4 motivated by above we determine the rank of a t gain unicyclic graph and classify the t gain graphs with rank 2

the two to infinity norm and singular subspace geometry with applications to high dimensional statistics the singular value matrix decomposition plays a ubiquitous role throughout statistics and related fields myriad applications including clustering classification and dimensionality reduction involve studying and exploiting the geometric structure of singular values and singular vectors this paper provides a novel collection of technical and theoretical tools for studying the geometry of singular subspaces using the two to infinity norm motivated by preliminary deterministic procrustes analysis we consider a general matrix perturbation setting in which we derive a new procrustean matrix decomposition together with flexible machinery developed for the two to infinity norm this allows us to conduct a refined analysis of the induced perturbation geometry with respect to the underlying singular vectors even in the presence of singular value multiplicity our analysis yields singular vector entrywise perturbation bounds for a range of popular matrix noise models each of which has a meaningful associated statistical inference task in addition we demonstrate how the two to infinity norm is the preferred norm in certain statistical settings specific applications discussed in this paper include covariance estimation singular subspace recovery and multiple graph inference both our procrustean matrix decomposition and the technical machinery developed for the two to infinity norm may be of independent interest

spectral statistics of bernoulli matrix ensembles a random walk approach i we investigate the eigenvalue statistics of random bernoulli matrices where the matrix elements are chosen independently from a binary set with equal probability this is achieved by initiating a discrete random walk process over the space of matrices and analysing the induced random motion of the eigenvalues an approach which is similar to dyson s brownian motion model but with important modifications in particular we show our process is described by a fokker planck equation up to an error margin which vanishes in the limit of large matrix dimension the stationary solution of which corresponds to the joint probability density function of certain well known fixed trace gaussian ensembles

signal plus noise matrix models eigenvector deviations and fluctuations summary estimating eigenvectors and low dimensional subspaces is of central importance for numerous problems in statistics computer science and applied mathematics in this paper we characterize the behaviour of perturbed eigenvectors for a range of signal plus noise matrix models encountered instatistical and random matrix theoretic settings we establish both first order approximation results i e sharp deviations and second order distributional limit theory i e fluctuations the concise methodology presented in this paper synthesizes tools rooted in two core concepts namely deterministic decompositions of matrix perturbations and probabilistic matrix concentration phenomena we illustrate our theoretical results with simulation examples involving stochastic block model random graphs

root locii for systems defined on hilbert spaces the root locus is an important tool for analyzing the stability and time constants of linear finite dimensional systems as a parameter often the gain is varied however many systems are modelled by partial differential equations or delay equations these systems evolve on an infinite dimensional space and their transfer functions are not rational in this paper a rigorous definition of the root locus for infinite dimensional systems is given and it is shown that the root locus is well defined for a large class of infinite dimensional systems as for finite dimensional systems any limit point of a branch of the root locus is a zero however the asymptotic behavior can be quite different from that for finite dimensional systems this point is illustrated with a number of examples it is shown that the familiar pole zero interlacing property for collocated systems with a hermitian state matrix extends to infinite dimensional systems with self adjoint generator this interlacing property is also shown to hold for collocated systems with a skew adjoint generator

graphs whose distance matrix has at most three negative eigenvalues abstract let d g be the distance matrix of a connected simple graph g the negative inertia of d g denoted by n d g is the number of negative eigenvalues of d g in this paper we determine all connected graphs g whose distance matrix d g has at most three negative eigenvalues

bounds of graph energy in terms of vertex cover number abstract the energy e g of a graph g is the sum of the absolute values of all eigenvalues of g in this paper we give a lower bound and an upper bound for graph energy in terms of vertex cover number for a graph g with vertex cover number it is proved that 2 2 c e g 2 where c is the number of odd cycles in g and is the maximum vertex degree of g the lower bound is attained if and only if g is the disjoint union of some complete bipartite graphs with perfect matchings and some isolated vertices the upper bound is attained if and only if g is the disjoint union of copies of k 1 together with some isolated vertices

the isotropic semicircle law and deformation of wigner matrices we analyze the spectrum of additive finite rank deformations of n n wigner matrices h the spectrum of the deformed matrix undergoes a transition associated with the creation or annihilation of an outlier when an eigenvalue di of the deformation crosses a critical value 1 this transition happens on the scale we allow the eigenvalues di of the deformation to depend on n under the condition we make no assumptions on the eigenvectors of the deformation in the limit n we identify the law of the outliers and prove that the nonoutliers close to the spectral edge have a universal distribution coinciding with that of the extremal eigenvalues of a gaussian matrix ensemble a key ingredient in our proof is the isotropic local semicircle law which establishes optimal high probability bounds on where m z is the stieltjes transform of wigner s semicircle law and v w are arbitrary deterministic vectors 2013 wiley periodicals inc

concentration of distances in wigner matrices abstract it is well known that distances in random iid matrices are highly concentrated around their mean in this note we extend this concentration phenomenon to wigner matrices exponential bounds for the lower tail are also included

root locus and boundary feedback design for a class of distributed parameter systems in this paper a fairly complete parallel of the finite dimensional root locus theory is presented for quite general nonconstant coefficient even order ordinary differential operators on a finite interval with control and output boundary conditions representative of a choice of colocated point actuators and sensors root locus design methods for linear distributed parameter systems have also been studied for some time and the primary difficulties in rigorously interpreting root locus conclusions for distributed parameter systems are well known first the transfer function of a distributed parameter system may not be meromorphic at infinity so that many of the standard rouche arguments required even in the lumped case to determine the asymptotic behavior of the root loci are not generally valid another difficulty is that the infinitesimal generator in the state space model for a closed loop system may not be selfadjoint accretive or even satisfy the spectrum determined growth condition thus regardless of whether the root loci interpreted as closed loop eigenvalues lie in the open left half plane additional analysis would be required to conclude that the closed loop system would be asymptotically stable formulating the systems in the classical format of a boundary control problem the asymptotic analysis of the root loci can be based on the pioneering work by birkhoff on eigenfunction expansions for boundary value problems work that predated and indeed motivated the development of spectral theory in hilbert space birkhoff s work also contains an asymptotic expansion of eigenfunctions in the spatial variable generalizing the earlier sturm liouville theory for second order operators by further extending this general asymptotic analysis to also include expansions in the gain parameter a rigorous treatment of the open and closed loop transfer functions and of the corresponding return difference equation can be presented the asymptotic analysis of the return difference equation forms the basis for both our rigorous formulation of the basic problem and its solution

the inertia of weighted unicyclic graphs let gw be a weighted graph the inertia of gw is the triple in gw i gw i gw i0 gw where i gw i gw i0 gw are the numbers of the positive negative and zero eigenvalues of the adjacency matrix a gw of gw including their multiplicities respectively i gw i gw are called the positive negative indices of inertia of gw respectively in this paper we present a lower bound for the positive negative indices of weighted unicyclic graphs of order n with fixed girth and characterize all weighted unicyclic graphs attaining this lower bound moreover we characterize the weighted unicyclic graphs of order n with two positive two negative and at least n 6 zero eigenvalues respectively

on the relationship between the skew rank of an oriented graph and the rank of its underlying graph abstract an oriented graph g is a digraph without loops and multiple arcs where g is the underlying graph of g let s g denote the skew adjacency matrix of g and a g be the adjacency matrix of g the rank resp skew rank of g resp g written as r g resp s r g refers to the rank of a g resp s g it is natural and interesting to study the relationship between s r g and r g wong ma and tian european j combin 54 2016 76 86 30 determined the sharp upper bound on s r g r g as a continuance of it in this paper a sharp lower bound on s r g r g is determined as well a sharp upper bound on s r g r g is determined all the corresponding extremal oriented graphs g are characterized respectively

on the nullity and the matching number of unicyclic graphs abstract let g be a graph with n vertices and g be the matching number of g let g denote the nullity of g the multiplicity of the eigenvalue zero of g it is well known that if g is a tree then g n 2 g tan and liu x tan b liu on the nullity of unicyclic graphs linear alg appl 408 2005 212 220 proved that the nullity set of all unicyclic graphs with n vertices is 0 1 n 4 and characterized the unicyclic graphs with g n 4 in this paper we characterize the unicyclic graphs with g n 5 and we prove that if g is a unicyclic graph then g equals n 2 g 1 n 2 g or n 2 g 2 we also give a characterization of these three types of graphs furthermore we determine the unicyclic graphs g with g 0 which answers affirmatively an open problem by tan and liu

eigenvalue statistics of the real ginibre ensemble the real ginibre ensemble consists of random n ifmmode times else texttimes fi n matrices formed from independent and identically distributed standard gaussian entries by using the method of skew orthogonal polynomials the general n point correlations for the real eigenvalues and for the complex eigenvalues are given as n ifmmode times else texttimes fi n pfaffians with explicit entries a computationally tractable formula for the cumulative probability density of the largest real eigenvalue is presented this is relevant to may s stability analysis of biological webs

lower bounds of graph energy in terms of matching number abstract the energy e g of a graph g is the sum of the absolute values of all eigenvalues of g we are interested in the relation between the energy of a graph g and the matching number g of g it is proved that e g 2 g for every graph g and e g 2 g 5 5 c 1 g if the cycles of g if any are pairwise vertex disjoint where c 1 g denotes the number of odd cycles in g besides we prove that e g r g 1 2 if g has at least one odd cycle and it is not of full rank where r g is the rank of g

spectral statistics of erdős renyi graphs i local semicircle law we consider the ensemble of adjacency matrices of erd s renyi random graphs that is graphs on n vertices where every edge is chosen independently and with probability p p n we rescale the matrix so that its bulk eigenvalues are of order one we prove that as long as pn with a speed at least logarithmic in n the density of eigenvalues of the erd s renyi ensemble is given by the wigner semicircle law for spectral windows of length larger than n 1 up to logarithmic corrections as a consequence all eigenvectors are proved to be completely delocalized in the sense that the l norms of the l2 normalized eigenvectors are at most of order n 1 2 with a very high probability the estimates in this paper will be used in the companion paper spectral statistics of erd s renyi graphs ii eigenvalue spacing and the extreme eigenvalues 2011 preprint to prove the universality of eigenvalue distributions both in the bulk and at the spectral edges under the further restriction that pn n2 3

a characterization of graphs with rank 4 the rank of a graph g is defined to be the rank of its adjacency matrix in this paper we consider the following problem what is the structure of a connected graph with rank 4 this question has not yet been fully answered in the literature and only some partial results are known in this paper we resolve this question by completely characterizing graphs g whose adjacency matrix has rank 4

energy matching number and odd cycles of graphs abstract the energy e g of a graph g is the sum of the absolute values of the eigenvalues of the adjacency matrix of g wong et al 2018 11 proved that e g 2 g where g denotes the matching number of g we settle the equality case that was left open from that paper we also show that for connected graphs without perfect matching e g 2 g 1 except for complete bipartite graphs of the form k r r 1 further for graphs with vertex disjoint cycles we prove that e g 2 g c o g where c o g denotes the number of odd cycles of g with length at least 5 this improves another result of wong et al

transfer functions for infinite dimensional systems in this paper we study three definitions of the transfer function for an infinite dimensional system the first one defines the transfer function as the expression c si a 1 b d in the second definition the transfer function is defined as the quotient of the laplace transform of the output and input with initial condition zero in the third definition we introduce the transfer function as the quotient of the input and output when the input and output are exponentials we show that these definitions always agree on the right half plane bounded to the left by the growth bound of the underlying semigroup but that they may differ elsewhere

smallest singular value of a random rectangular matrix we prove an optimal estimate of the smallest singular value of a random sub gaussian matrix valid for all dimensions for an nn matrix a with inde pendent and identically distributed sub gaussian entries the smallest singular value of a is at least of the order p np n 1 with high probability a sharp

random matrices close to hermitian or unitary overview of methods and results the paper discusses recent progress in understanding statistical properties of eigenvalues of weakly non hermitian and non unitary random matrices the first type of ensembles is of the form i with being a large random n n hermitian matrix with independent entries deformed by a certain anti hermitian n n matrix i satisfying in the limit of large dimension n the condition tr 2 n tr 2 here can be either a random or just a fixed given hermitian matrix ensembles of such a type with 0 emerge naturally when describing quantum scattering in systems with chaotic dynamics and serve to describe resonance statistics related models are used to mimic complex spectra of the dirac operator with chemical potential in the context of quantum chromodynamics ensembles of the second type arising naturally in scattering theory of discrete time systems are formed by n n matrices with complex entries such that i for 0 this coincides with the circular unitary ensemble and 0 i describes deviation from unitarity our result amounts to answering statistically the following old question given the singular values of a matrix describe the locus of its eigenvalues we systematically show that the obtained expressions for the correlation functions of complex eigenvalues describe a non trivial crossover from wigner dyson statistics of real unimodular eigenvalues typical of hermitian unitary matrices to ginibre statistics in the complex plane typical of ensembles with strong non hermiticity tr 2 tr 2 when n finally we discuss scarce results available on eigenvector statistics for weakly non hermitian random matrices

on the nullity of bicyclic graphs the nullity of a graph is the multiplicity of the eigenvalue zero in its spectrum in this paper we obtain the nullity set of n vertex unicyclic graphs and characterize the unicyclic graphs with extremal nullity

per spectral characterizations of graphs with extremal per nullity abstract a graph g is said to be determined by its permanental spectrum if any graph having the same permanental spectrum as g is isomorphic to g in this paper we introduce the permanental nullity of a graph the multiplicity of the number zero in the permanental spectrum of a graph to study graphs determined by their permanental spectra first we determine all graphs of order n whose permanental nullities are n 2 n 3 n 4 and n 5 respectively then we show that all graphs with the permanental nullity n 2 n 3 or n 5 and all non bipartite graphs with the permanental nullity n 4 are determined by their permanental spectra in particular we prove that the complete bipartite graphs are determined by their permanental spectra

lower bounds for the energy of digraphs abstract the energy of a digraph d is defined as e d i 1 n re z i where z 1 z n are the eigenvalues of d in this article we find lower bounds for the energy of digraphs in terms of the number of closed walks of length 2 extending in this way the result obtained by caporossi et al g caporossi d cvetkovic i gutman p hansen variable neighborhood search for extremal graphs 2 finding graphs with extremal energy j chem inf comput sci 39 1999 984 996 2 m e g for all graphs g with m edges also we study digraphs with three eigenvalues

random matrices universality of esds and the circular law given an n x n complex matrix a let mu a x y 1 n vertical bar 1 i n re lambda i x im lambda i y vertical bar be the empirical spectral distribution esd of its eigenvalues lambda i is an element of c i l n we consider the limiting distribution both in probability and in the almost sure convergence sense of the normalized esd mu 1 root n an of a random matrix a n a ij 1 i j n where the random variables a ij e a ij are i i d copies of a fixed random variable x with unit variance we prove a universality principle for such ensembles namely that the limit distribution in question is independent of the actual choice of x in particular in order to compute this distribution one can assume that x is real or complex gaussian as a related result we show how laws for this esd follow from laws for the singular value distribution of 1 root n a n zi for complex z as a corollary we establish the circular law conjecture both almost surely and in probability which asserts that mu 1 root n an converges to the uniform measure on the unit disc when the a ij have zero mean

a characterization of long graphs of arbitrary rank abstract the rank of a graph g is defined to be the rank of its adjacency matrix a connected graph g of rank n is called long if it has maximum diameter among all connected graphs with rank n in this paper we characterize the long graphs of rank n for n an arbitrary positive integer

the energy of a graph abstract the energy e g of a simple graph g is defined to be the sum of the absolute values of the eigen values of g if g is a k regular graph on n vertices then e g k k n 1 n k b 2 and this bound is sharp it is shown that for each 0 there exist infinitely many n for each of which there exists a k regular graph g of order n with k e g b 2 two graphs with the same number of vertices are equienergetic if they have the same energy we show that for any positive integer n 3 there exist two equienergetic graphs of order 4n that are not cospectral

on the nullity of graphs with pendent vertices abstract the nullity of a graph g denoted by g is the multiplicity of the eigenvalue zero in its spectrum cheng and liu b cheng b liu on the nullity of graphs electron j linear algebra 16 2007 60 67 characterized the extremal graphs attaining the upper bound n 2 and the second upper bound n 3 in this paper as the continuance of it we determine the extremal graphs with pendent vertices achieving the third upper bound n 4 and fourth upper bound n 5 we then proceed recursively to construct all graphs with pendent vertices which satisfy g 0 our results provide a unified approach to determine n vertex unicyclic respectively bicyclic and tricyclic graphs which achieve the maximal and second maximal nullity and characterize n vertex extremal trees attaining the second and third maximal nullity as a consequence we respectively determine the nullity sets of trees unicyclic graphs bicyclic graphs and tricyclic graphs on n vertices

second order systems with acceleration measurements accelerometers are often used to measure the output of second order systems such as structural vibrations conditions under which these systems are well posed are obtained we also establish conditions under which these systems have minimum phase transfer functions

on delocalization of eigenvectors of random non hermitian matrices we study delocalization of null vectors and eigenvectors of random matrices with i i d entries let a be an n times n random matrix with i i d real subgaussian entries of zero mean and unit variance we show that with probability at least 1 e log 2 n begin aligned min limits i subset n i m vert mathbf v i vert ge frac m 3 2 n 3 2 log cn vert mathbf v vert end aligned for any real eigenvector mathbf v and any m in log c n n where mathbf v i denotes the restriction of mathbf v to i further when the entries of a are complex with i i d real and imaginary parts we show that with probability at least 1 e log 2 n all eigenvectors of a are delocalized in the sense that begin aligned min limits i subset n i m vert mathbf v i vert ge frac m n log cn vert mathbf v vert end aligned for all m in log c n n comparing with related results in the range m in log c n n log c n in the i i d setting and with weaker probability estimates our lower bounds on vert mathbf v i vert strengthen an earlier estimate min nolimits i m vert mathbf v i vert ge c m n 6 vert mathbf v vert obtained in rudelson and vershynin geom funct anal 26 6 1716 1776 2016 and bounds min nolimits i m vert mathbf v i vert ge c m n 2 vert mathbf v vert in the real setting and min nolimits i m vert mathbf v i vert ge c m n 3 2 vert mathbf v vert in the complex setting established in luh and o rourke eigenvector delocalization for non hermitian random matrices and applications arxiv 1810 00489 as the case of real and complex gaussian matrices shows our bounds are optimal up to the polylogarithmic multiples we derive stronger estimates without the polylogarithmic error multiples for null vectors of real n 1 times n random matrices

induced ginibre ensemble of random matrices and quantum operations a generalization of the ginibre ensemble of non hermitian random square matrices is introduced the corresponding probability measure is induced by the ensemble of rectangular gaussian matrices via a quadratization procedure we derive the joint probability density of eigenvalues for such an induced ginibre ensemble and study various spectral correlation functions for complex and real matrices and analyse universal behaviour in the limit of large dimensions in this limit the eigenvalues of the induced ginibre ensemble cover uniformly a ring in the complex plane the real induced ginibre ensemble is shown to be useful to describe the statistical properties of evolution operators associated with random quantum operations for which the dimensions of the input state and the output state do differ

bounds for the positive and negative inertia index of a graph abstract let g be a graph and let a g be adjacency matrix of g the positive inertia index respectively the negative inertia index of g denoted by p g respectively n g is defined to be the number of positive eigenvalues respectively negative eigenvalues of a g in this paper we present the bounds for p g and n g as follows m g c g p g m g c g m g c g n g m g c g where m g and c g are respectively the matching number and the cyclomatic number of g furthermore we characterize the graphs which attain the upper bounds and the lower bounds respectively

graph energy change due to edge deletion the energy of a graph is the sum of the singular values of its adjacency matrix we are interested in how the energy of a graph changes when edges are deleted examples show that all cases are possible increased decreased unchanged our goal is to find possible graph theoretical descriptions and to provide an infinite family of graphs for each case the main tool is a singular value inequality for complementary submatrices and its equality case

spectral statistics of erdős renyi graphs ii eigenvalue spacing and the extreme eigenvalues we consider the ensemble of adjacency matrices of erd s renyi random graphs i e graphs on n vertices where every edge is chosen independently and with probability p p n we rescale the matrix so that its bulk eigenvalues are of order one under the assumption p n gg n 2 3 we prove the universality of eigenvalue distributions both in the bulk and at the edge of the spectrum more precisely we prove 1 that the eigenvalue spacing of the erd s renyi graph in the bulk of the spectrum has the same distribution as that of the gaussian orthogonal ensemble and 2 that the second largest eigenvalue of the erd s renyi graph has the same distribution as the largest eigenvalue of the gaussian orthogonal ensemble as an application of our method we prove the bulk universality of generalized wigner matrices under the assumption that the matrix entries have at least 4 e moments

on the characterization of digraphs with given rank abstract the rank of a digraph is defined as the rank of its adjacency matrix in this paper we show how to find all digraphs of rank k from the reduced bipartite graphs of rank 2k in particular since the bipartite graphs of rank 2 4 and 6 are known then we characterize digraphs of rank 1 2 and 3 the results are based on rank preserving operations on digraphs such as splitting and fusion of vertices or twin deletion and vertex multiplication

statistics of real eigenvalues in ginibre s ensemble of random real matrices the integrable structure of ginibre s orthogonal ensemble of random matrices is looked at through the prism of the probability p n k to find exactly k real eigenvalues in the spectrum of an n ifmmode times else texttimes fi n real asymmetric gaussian random matrix the exact solution for the probability function p n k is presented and its remarkable connection to the theory of symmetric functions is revealed an extension of the dyson integration theorem is a key ingredient of the theory presented

on nonconvergence of adjoint semigroups for control systems with delays it is shown that the adjoints of a spline based approximation scheme for delay equations do not converge strongly

mixed graphs with h rank 3 abstract in the article mohar 2016 9 mohar determined all mixed graphs with h rank 2 and used it to classify cospectral graphs with respect to their hermitian adjacency matrix constructing a class of graphs which can not be determined by their h spectrum in the present paper we investigate the h rank of mixed graphs further determining the h ranks of those mixed graphs with trees cycles and complete bipartite graphs as underlying graphs respectively moreover we characterize all mixed graphs with h rank 3 and show that all connected mixed graphs with h rank 3 can be determined by their h spectrum

upper bound for intermediate singular values of random matrices abstract in this paper we prove that an n n matrix a with independent centered subgaussian entries satisfies s n 1 l a c 1 t l n with probability at least 1 exp c 2 t l this yields s n 1 l a c l n in combination with a known lower bound these results can be generalized to the rectangular matrix case

delocalization of eigenvectors of random matrices with independent entries author s rudelson mark vershynin roman abstract we prove that an n by n random matrix g with independent entries is completely delocalized suppose the entries of g have zero means variances uniformly bounded below and a uniform tail decay of exponential type then with high probability all unit eigenvectors of g have all coordinates of magnitude o n 1 2 modulo logarithmic corrections this comes a consequence of a new geometric approach to delocalization for random matrices

localization and delocalization of eigenvectors for heavy tailed random matrices consider an n times n hermitian random matrix with above the diagonal independent entries with alpha stable symmetric distribution and 0 2 we give vanishing bounds on the l p norm of the eigenvectors normalized to have unit l 2 norm on the contrary when 0 alpha 2 3 we prove that these eigenvectors are localized

the positive and the negative inertia index of line graphs of trees abstract the positive resp negative inertia index of g denoted by p g resp n g is defined to be the number of positive resp negative eigenvalues of its adjacency matrix the nullity of g denoted by g is defined to be the multiplicity of the eigenvalue zero in the adjacency spectrum of g in 2001 gutman and sciriha 11 proved that the nullity of the line graph of a tree is either 0 or 1 in this paper we consider the positive and the negative inertia index for the line graph l t of a tree t showing that e t 1 2 p l t e t 1 where e t denotes the number of internal non pendant edges contained in t the extremal trees for which p l t attains the upper bound and the lower bound are respectively characterized it is shown that l t is nonsingular if p l t attains the upper bound and l t is singular if p l t attains the lower bound

minimum phase infinite dimensional second order systems in general better performance can be achieved with a controlled minimum phase system than a controlled nonminimum phase system we show that a wide class of second order infinite dimensional systems with either velocity or position measurements are minimum phase the results are illustrated by two examples

graphs whose adjacency matrices have rank equal to the number of distinct nonzero rows abstract for a simple graph g let rank g and dnzr g denote respectively the rank and the number of distinct nonzero rows of the adjacency matrix a g of g equivalent conditions are given for the join g 1 g 2 of two vertex disjoint graphs g 1 g 2 to satisfy rank g 1 g 2 dnzr g 1 g 2 a new proof is provided for the known relation rank g dnzr g for cographs g our approach relies on the concepts of neighborhood equivalence classes reduced graph and reduced adjacency matrix and also on a known result that relates the spectrum of the adjacency matrix of a graph with that of its reduced adjacency matrix as well as a new characterization of the nonsingularity of a real symmetric matrix in a special 2 2 block form our treatment provides ways to construct graphs g other than cographs that satisfy rank g dnzr g as a side result we also show that every rational number is equal to the sum of the entries of the inverse of the adjacency matrix of a connected nonsingular graph