abrikosov lattices in finite domains in 1957 abrikosov published his work on periodic solutions to the linearized ginzburg landau equations abrikosov s analysis assumes periodic boundary conditions which are very different from the natural boundary conditions the minimizer of the ginzburg landau energy functional should satisfy in the present work we prove that the global minimizer of the fully non linear functional can be approximated in every rectangular subset of the domain by one of the periodic solution to the linearized ginzburg landau equations in the plane furthermore we prove that the energy of this solution is close to the minimum of the energy over all abrikosov s solutions in that rectangle

symmetric vortices for the ginzberg landau equations of superconductivity and the nonlinear desingularization phenomenon abstract the existence of countably many distinct symmetric vortices for the nonlinear ginzberg landau equations of superconductivity with an arbitrary positive parameter is proven we then prove the existence of a linearization phenomenon as the parameter in this case the linearized equations contain an appropriate dirac delta function these equations are known as the london equations

edge connectivity in regular multigraphs from eigenvalues abstract let g be a d regular multigraph and let 2 g be the second largest eigenvalue of g in this paper we prove that if 2 g d 1 9 d 2 10 d 17 4 then g is 2 edge connected furthermore for t 2 we show that g is t 1 edge connected when 2 g d t and in fact when 2 g d t 1 if t is odd

geometric invariant theory and the generalized eigenvalue problem let g be a connected reductive subgroup of a complex connected reductive group hat g fix maximal tori and borel subgroups of g and hat g consider the cone mathcal lr g hat g generated by the pairs nu hat nu of dominant characters such that v nu is a submodule of v hat nu with usual notation here we give a minimal set of inequalities describing mathcal lr g hat g as a part of the dominant chamber in other words we describe the facets of mathcal lr g hat g which intersect the interior of the dominant chamber we also describe smaller faces finally we are interested in some classical redundant inequalities

eigenvalues invariant factors highest weights and schubert calculus we describe recent work of klyachko totaro knutson and tao that characterizes eigenvalues of sums of hermitian matrices and decomposition of tensor products of representations of gln c we explain related applications to invariant factors of products of matrices intersections in grassmann varieties and singular values of sums and products of arbitrary matrices recent breakthroughs primarily by a klyachko b totaro a knutson and t tao with contributions by p belkale and c woodward have led to complete solutions of several old problems involving the various notions in the title our aim here is to describe this work and especially to show how these solutions are derived from it along the way we will see that these problems are also related to other areas of mathematics including geometric invariant theory symplectic geometry and combinatorics in addition we present some related applications to singular values of arbitrary matrices although many of the theorems we state here have not appeared elsewhere their proofs are mostly soft algebra based on the hard geometric or combinatorial work of others indeed this paper emphasizes concrete elementary arguments we do give some new examples and counterexamples and raise some new open questions we have attempted to point to the sources and to some of the key partial results that had been conjectured or proved before however there is a very large literature particularly for linear algebra problems about eigenvalues singular values and invariant factors we have listed only a few of these articles from whose bibliographies we hope an interested reader can trace the history we apologize to the many whose work is not cited directly we begin in the first five sections by describing each of the problems together with some of their early histories and we state as theorems the new solutions to these problems in section 6 we describe the steps toward these solutions that were carried out before the recent breakthroughs then we discuss the recent solutions and explain how these theorems follow from the work of the above mathematicians sections 7 8 9 and 10 also contain variations and generalizations of some of the theorems stated in the first five sections as well as attributions of the theorems to their authors one of our fascinations with this subject even now that we have proofs of the theorems is the challenge to understand in a deeper way why all these subjects are received by the editors in july 1999 and in revised form january 3 2000 2000 mathematics subject classification primary 15a42 22e46 14m15 secondary 05e15 13f10 14c17 15a18 47b07 the author was partly supported by nsf grant dms9970435 c 2000 american mathematical society

edge disjoint spanning trees and eigenvalues of regular graphs abstract partially answering a question of paul seymour we obtain a sufficient eigenvalue condition for the existence of k edge disjoint spanning trees in a regular graph when k 2 3 more precisely we show that if the second largest eigenvalue of a d regular graph g is less than d 2 k 1 d 1 then g contains at least k edge disjoint spanning trees when k 2 3 we construct examples of graphs that show our bounds are essentially best possible we conjecture that the above statement is true for any k d 2

vertex connectivity and eigenvalues of graphs abstract let g n 1 g 2 g and q 2 g denote the vertex connectivity the algebraic connectivity the second largest adjacency eigenvalue and the second largest signless laplacian eigenvalue of g respectively in this paper we prove that for an integer k 0 and any simple graph g of order n with maximum degree and minimum degree k the vertex connectivity g k if n 1 g h 2 k or 2 g h 2 k or q 2 g 2 h 2 k where h 2 k k 1 n n k 1 k 1 4 k 2 n 1 which improves the result in appl math comput 344 345 2019 141 149 and the result in electron j linear algebra 34 2018 428 443 analogue results involving n 1 g 2 g and q 2 g to characterize vertex connectivity of regular graphs triangle free graphs and graphs with fixed girth are also presented

computations of the first eigenpairs for the schrodinger operator with magnetic field this paper is devoted to computations of eigenvalues and eigenvectors for the schrodinger operator with constant magnetic field in a domain with corners as the semi classical parameter h tends to 0 the eigenvectors corresponding to the smallest eigenvalues concentrate in the corners they have a two scale structure consisting of a corner layer at scale sqrt h and an oscillatory term at scale h the high frequency oscillations make the numerical computations particularly delicate we propose a high order finite element method to overcome this difficulty relying on such a discretization we illustrate theoretical results on plane sectors squares and other straight or curved polygons we conclude by discussing convergence issues

magnetic wkb constructions this paper is devoted to the semiclassical magnetic laplacian until now wkb expansions for the eigenfunctions were only established in the presence of a non zero electric potential here we tackle the pure magnetic case thanks to feynman hellmann type formulas and coherent states decomposition we develop here a magnetic born oppenheimer theory exploiting the multiple scales of the problem we are led to solve an effective eikonal equation in pure magnetic cases and to obtain wkb expansions we also investigate explicit examples for which we can improve our general theorem global wkb expansions quasi optimal agmon estimates and upper bound of the tunelling effect in symmetric cases we also apply our strategy to get more accurate descriptions of the eigenvalues and eigenfunctions in a wide range of situations analyzed in the last two decades

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

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

edge disjoint spanning trees and eigenvalues of graphs it is proven that a graph of order n with minimum degree 2k has k edge disjoint spanning trees if its second largest eigenvalue is less than 2k 1 1 o 1 n this extends earlier results of cioab and wong and improves results of gu lai li and yao and also confirms approximately their conjectures for sufficiently large graphs with respect to the minor term we also conjecture that the minor term can be removed

peak power in the 3d magnetic schrodinger equation this paper is devoted to the spectral analysis of the magnetic neumann laplacian on an infinite cone of aperture alpha when the magnetic field is constant and parallel to the revolution axis and when the aperture goes to zero we prove that the first n eigenvalues exist and admit asymptotic expansions in powers of alpha 2

a generalization of fulton s conjecture for arbitrary groups we prove a generalization of fulton s conjecture which relates intersection theory on an arbitrary flag variety to invariant theory

surface superconductivity in 3 dimensions we study the ginzburg landau system for a superconductor occupying a 3 dimensional bounded domain and improve the estimate of the upper critical field h c3 obtained by k lu and x pan in j diff eqns 168 2000 386 452 we also analyze the behavior of the order parameters we show that under an applied magnetic field lying below and not far from h c3 order parameters concentrate in a vicinity of a sheath of the surface that is tangential to the applied field and exponentially decay both in the normal and tangential directions away from the sheath in the l 2 sense as the applied field decreases further but keeps in between and away from h c2 and h c3 the superconducting sheath expands but does not cover the entire surface and superconductivity at the surface portion orthogonal to the applied field is always very weak this phenomenon is significantly different to the surface superconductivity on a cylinder of infinite height studied by x pan in comm math phys 228 2002 327 370 where under an axial applied field lying in between h c2 and h c3 the entire surface is in the superconducting state

stochastic stability of continuous time consensus protocols a unified approach to studying convergence and stochastic stability of continuous time consensus protocols cps is presented in this work our method applies to networks with directed information flow both cooperative and noncooperative interactions networks under weak stochastic forcing and those whose topology and strength of connections may vary in time the graph theoretic interpretation of the analytical results is emphasized we show how the spectral properties such as algebraic connectivity and total effective resistance as well as the geometric properties such as the dimension and the structure of the cycle subspace of the underlying graph shape stability of the corresponding cps in addition we explore certain implications of spectral graph theory to cp design in particular we point out that expanders sparse highly connected graphs generate cps whose performance remains uniformly high when the size of the network grows unboundedly similarly we highlight the benefits of using random

analytic description of layer undulations in smectic a liquid crystals we investigate the layer undulations that appear in smectic a liquid crystals when a magnetic field is applied in the direction parallel to the smectic layers in an earlier work garcia cervera and joo in j comput theor nanosci 7 795 801 2010 the authors characterized the critical field using the landau de gennes model for smectic a liquid crystals in this paper we obtain an asymptotic expression of the unstable modes using convergence theory and a sharp estimate of the critical field under the assumption that the layers are fixed at the boundaries the maximum layer undulation occurs in the middle of the cell and the displacement amplitude decreases near the boundaries our estimate of the critical field is consistent with the helfrich hurault theory when natural boundary conditions are considered the displacement amplitude does not diminish near the boundary in sharp contrast with the dirichlet case and the critical field is reduced compared to the one calculated in the classical theory this is consistent with the experiments carried out by ishikawa and lavrentovich phys rev e 63 030501 r 2001 furthermore we prove the existence and stability of the solution to the nonlinear system of the landau de gennes model using bifurcation theory numerical simulations are used to illustrate the predictions of the analysis

note on edge disjoint spanning trees and eigenvalues abstract let g 2 g n 1 g and 2 g be the maximum number of edge disjoint spanning trees the second largest adjacency eigenvalue the algebraic connectivity and the second largest signless laplace eigenvalue of g respectively in this note we prove that for any graph g with minimum degree 2 k if 2 g 2 k 1 1 or n 1 g 2 k 1 1 or 2 g 2 2 k 1 1 then g k which confirms a conjecture of liu hong and lai and also implies a conjecture of cioab and wong

analysis and approximation of the ginzburg landau model of superconductivity the authors consider the ginzburg landau model for superconductivity first some well known features of superconducting materials are reviewed and then various results concerning the model the resultant differential equations and their solution on bounded domains are derived then finite element approximations of the solutions of the ginzburg landau equations are considered and error estimates of optimal order are derived

the phase transitions from chiral nematic toward smectic liquid crystals a chen lubensky energy is used to investigate phase transitions from chiral nematic to smectic c and smectic a liquid crystal phases we consider a liquid crystalline material confined between two parallel plates where the dimensions of are assumed to be large relative both to the width of a smectic layer and the material s chiral pitch we take boundary conditions so that the smectic phase melts at the plates surfaces and prove the existence of energy minimizers in an admissible set consisting of order parameters psi in mathcal h 2 0 omega and molecular directors mathbf n in mathbf w 1 2 omega mathbb s 2 then under the physically observed assumption that the frank elasticity constants become large near a phase transition we establish estimates for the transition region separating phases in particular we derive analytic estimates proving that chirality lowers the transition temperature regime above which minimizers are nematic and below which minimizers are in a smectic phase

star complements and edge connectivity in finite graphs abstract let g be a finite graph with h as a star complement for an eigenvalue other than 0 or 1 let g g denote respectively the vertex connectivity and minimum degree of g we prove that g is controlled by g and h in particular for each k n there exists a smallest non negative integer f k such that g k whenever h k and g f k we show that f 1 0 f 2 2 f 3 3 f 4 5 and f 5 7

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

emergence of the triangular lattice near a flat wall the bifurcation of periodic solutions near a flat wall for applied magnetic fields which are slightly weaker than hc2 is considered for a reduced ginzburg landau model obtained in the large limit we formally demonstrate that following the bifurcation of the first mode when the applied magnetic field is further decreased there is a second bifurcation after which the solution develops continuously into the well known triangular lattice

edge disjoint spanning trees and eigenvalues abstract let g and 2 g be the maximum number of edge disjoint spanning trees and the second largest eigenvalue of a graph g respectively motivated by a question of seymour on the relationship between eigenvalues of a graph g and g cioab and wong conjectured that for any integers k 2 d 2 k and a d regular graph g if 2 g d 2 k 1 d 1 then g k they proved this conjecture for k 2 3 gu lai li and yao generalized this conjecture to simple graph and conjectured that for any integer k 2 and a graph g with minimum degree and maximum degree if 2 g 2 2 k 1 1 then g k in this paper we prove that 2 g 2 k 2 k 1 implies g k and show the two conjectures hold for sufficiently large n

homotopy classification of minimizers of the ginzburg landau energy and the existence of permanent currents our objective is to explain the phenomenon of permanent currents within the context of the ginzburg landau model for superconductors using variational techniques we make a connection between the formation of permanent currents and the topology of the superconducting sample

on graphs with equal algebraic and vertex connectivity let g be an undirected unweighted graph on n vertices let l be its laplacian matrix and let l l i j be the group inverse of l it is known that for z l 1 2 max1 i j n s 1n li s lj s the quantity 1 z l is a lower bound on the algebraic connectivity a g of g while the vertex connectivity of g v g is an upper bound on a g we characterize the graphs g for which v g a g and subsequently prove that if n v g 2 then v g a g holds if and only if 1 z l a g v g we close with an example showing that the equality 1 z l a g does not necessarily imply that 1 z l a g v g

edge connectivity and signless laplacian eigenvalue of graphs in this paper we first show that if the second smallest laplacian eigenvalue of a graph is no less than k 1 n 1 n 1 or the second largest signless laplacian eigenvalue of a graph is no more than 2 k 1 n 1 n 1 then the graph is k edge connected where is the minimum degree of the graph and n is the order of the graph also we give a laplacian eigenvalue condition and a signless laplacian eigenvalue condition for a graph to be k edge connected involving the girth g of the graph respectively our results generalize some known results

eigenvalues and edge connectivity of regular graphs abstract in this paper we show that if the second largest eigenvalue of a d regular graph is less than d 2 k 1 d 1 then the graph is k edge connected when k is 2 or 3 we prove stronger results let d denote the largest root of x 3 d 3 x 2 3 d 2 x 2 0 we show that if the second largest eigenvalue of a d regular graph g is less than d then g is 2 edge connected and we prove that if the second largest eigenvalue of g is less than d 3 d 3 2 16 2 then g is 3 edge connected

the onset of superconductivity in semi infinite strips the existence of a superconducting branch bifurcating from the normal state is proved in semi infinite strips it is proved that the critical magnetic field at which bifurcation takes place or the onset field for a semi infinite strip is greater than the onset field for an infinite strip with the same width in addition we find the loci of the vortices far away from the corners and show convergence of the bifurcating modes in long rectangles to those in the semi infinite strip with the same width

numerical approximations of the ginzburg landau models for superconductivity in this paper we review various methods for the numerical approximations of the ginzburg landau models of superconductivity particular attention is given to the different treatment of gauge invariance in both the finite element finite difference and finite volume settings representative theoretical results typical numerical simulations and computational challenges are presented generalizations to other relevant models are also discussed

spanning tree packing number and eigenvalues of graphs with given girth abstract let g and g denote the spanning tree packing number and the edge connectivity of a graph g respectively cioab and wong 2012 in 5 conjectured an explicit relationship between g and the second largest adjacency eigenvalue 2 g of a regular graph gu et al 2016 in 12 presented a more general conjecture on a simple graph g this conjecture was proved by liu et al 2014 in 21 by showing that for any simple graph g with minimum degree 2 k 4 if 2 g 2 k 1 1 then g k similar results involving the algebraic connectivity n 1 g and the second largest signless laplacian eigenvalue q 2 g of a graph g were also obtained in this paper we determine a moore function f g for a graph g with minimum degree and girth g and prove that if g is a simple graph of order n with minimum degree 2 k 4 and girth g then i if 2 g 2 k 1 f g then g k ii if n 1 g 2 k 1 f g then g k iii if q 2 g 2 2 k 1 f g then g k the edge connectivity analogue results are also obtained former results in gu et al 2016 12 li et al 2013 18 liu et al 2014 20 and liu et al 2014 21 are extended

extremal rays in the hermitian eigenvalue problem the hermitian eigenvalue problem asks for the possible eigenvalues of a sum of n times n hermitian matrices given the eigenvalues of the summands the regular faces of the cones gamma n s controlling this problem have been characterized in terms of classical schubert calculus by the work of several authors we determine extremal rays of gamma n s which are never regular faces by relating them to the geometry of flag varieties the extremal rays either arise from modular intersection loci or by induction from extremal rays of smaller groups explicit formulas are given for both the extremal rays coming from such intersection loci and for the induction maps

spectral conditions for edge connectivity and packing spanning trees in multigraphs abstract a multigraph is a graph with possible multiple edges but no loops the multiplicity of a multigraph is the maximum number of edges between any pair of vertices we prove that for a multigraph g with multiplicity m and minimum degree 2 k if the algebraic connectivity is greater than min 2 k 1 1 m 2 k 1 2 then g has at least k edge disjoint spanning trees for a multigraph g with multiplicity m and minimum degree k if the algebraic connectivity is greater than min 2 k 1 1 m k 1 then the edge connectivity is at least k these extend some earlier results a balloon of a graph g is a maximal 2 edge connected subgraph that is joined to the rest of g by exactly one cut edge we provide spectral conditions for the number of balloons in a multigraph which also generalizes an earlier result

remarks on the spectrum of the neumann problem with magnetic field in the half space we consider a schrodinger operator with a constant magnetic field in a one half three dimensional space with neumann type boundary conditions it is known from the works by lu pan and helffer morame that the lower bound of its spectrum is less than b the intensity of the magnetic field provided that the magnetic field is not normal to the boundary we prove that the spectrum under b is a finite set of eigenvalues each of infinite multiplicity in the case when the angle between the magnetic field and the boundary is small we give a sharp asymptotic expansion of the number of these eigenvalues

radial solutions for the ginzburg landau equation with applied magnetic field abstract the purpose of this work is a systematic study of symmetric vortices for the ginzburg landau model of superconductivity along a cylinder with applied magnetic field parallel to its axis the ginzburg landau constant of the material and the degree d of the vortex are fixed for any given parameters r the radius of the cylinder and h the intensity of the applied magnetic field one can find a symmetric vortex a which satisfies the boundary conditions b r e i d and curl a b r he 3 it is then shown that symmetric vortices form a family depending continuously on two real parameters and c which describe the behaviour at the center of the vortex as the boundary conditions depend smoothly on those parameters one can distinguish two main connected domains of vortices the first one defined by the boundary conditions b r e i d and ra r b r is a zone of stability where remains increasing the second one defined by the boundary conditions b r e i d and ra r b r d is a zone where some instability can occur attention is focused on the smooth branch of vortices which separates those two domains it is indexed by the parameter running from some l 0 to and the limit as decreases to l corresponds to a berger and chen type vortex

stable nucleation for the ginzburg landau system with an applied magnetic field we consider the ginzburg landau system with an applied magnetic field and analyze the behavior of solutions when the domain is a cylinder of radius overline r and the applied field is parallel to the axis it is shown that there is an upper critical value overline h such that if the modulus of the applied field is greater than overline h the normal nonsuperconducting state in which the order parameter is identically zero is stable and if the modulus of the applied field is slightly below oh the normal state is unstable in addition it is shown that there is a positive lower critical value underline h leq overline h such that the normal state is unstable if the modulus of the applied field is less than underline h and stable if the modulus is slightly above underline h in the case of type ii materials for whic h the ginzburg landau constant is large it is shown that there is a discrete set of radii such that if overline r notin cal b kappa and kappa overline r is sufficiently large then for each applied field of modulus slightly less than overline h or slightly more than underline h there is precisely one small superconducting solution up to a gauge transformation which is stable moreover for this solution the complex valued order parameter is zero only on the axis of the cylinder and its winding number is proportional to the product of 2 and the cross sectional area of the cylinder in addition the solution exhibits surface superconductivity as predicted by the physicists de gennes and st james

schrodinger operators with non degenerately vanishing magnetic fields in bounded domains we establish an asymptotic estimate of the lowest eigenvalue bf of the schrodinger operator 2 bf with a magnetic field in a bounded 2 dimensional domain where curl f vanishes non degenerately and b is a large parameter our study is based on an analysis on an eigenvalue variation problem for the sturm liouville problem using the estimate we determine the value of the upper critical field for superconductors subject to non homogeneous applied magnetic fields and localize the nucleation of superconductivity

uniqueness of symmetric vortex solutions in the ginzburg landau model of superconductivity abstract symmetric vortices are finite energy solutions a to the ginzburg landau equations of superconductivity with the form f r e id a s r r 2 y x the existence regularity and asymptotic form of the solutions f r s r for any d z 0 have been established by plohr and by burger and chen in this paper we prove the uniqueness of these solutions when the ginzburg landau parameter satisfies 2 2 d 2 for any fixed d z 0 to do this we show that any such solution is a non degenerate relative minimizer of the free energy functional constrained to a convex set then use a version of the mountain pass theorem to derive a contradiction should there be more than one solution

spectral conditions for some graphical properties abstract by a unified approach we present sufficient conditions based on spectral radius for a graph to be k connected k edge connected k hamiltonian k edge hamiltonian deficient and k path coverable

magnetic bottles in connection with superconductivity abstract motivated by the theory of superconductivity and more precisely by the problem of the onset of superconductivity in dimension two a lot of papers devoted to the analysis in a semi classical regime of the lowest eigenvalue of the schrodinger operator with magnetic field have appeared recently here we mention the works by bernoff sternberg lu pan and del pino felmer sternberg this partially recovers questions analyzed in a different context by the authors around the question of the so called magnetic bottles our aim is to analyze the former results to treat them in a more systematic way and to improve them by giving sharper estimates of the remainder in particular we improve significantly the lower bounds and as a by product we solve a conjecture proposed by bernoff sternberg concerning the localization of the ground state inside the boundary in the case with constant magnetic fields

boundary concentration for eigenvalue problems related to the onset of superconductivity we examine the asymptotic behavior of the eigenvalue h and corresponding eigenfunction associated with the variational problem in the regime h 1 here a is any vector field with curl equal to 1 the problem arises within the ginzburg landau model for superconductivity with the function h yielding the relationship between the critical temperature vs applied magnetic field strength in the transition from normal to superconducting state in a thin mesoscopic sample with cross section omega subset r 2 we first carry out a rigorous analysis of the associated problem on a half plane and then rigorously justify some of the formal arguments of bs obtaining an expansion for while also proving that the first eigenfunction decays to zero somewhere along the sample boundary when is not a disc for interior decay we demonstrate that the rate is exponential

connectivity and eigenvalues of graphs with given girth or clique number abstract let g n 1 g and 1 g denote the edge connectivity the algebraic connectivity and the laplacian spectral radius of g respectively in this paper we prove that for integers k 2 and r 2 and any simple graph g of order n with minimum degree k girth g 3 and clique number g r the edge connectivity g k if n 1 g k 1 n n g n n g or if n 1 g k 1 n r n r where n g is the moore bound on the smallest possible number of vertices such that there exists a regular simple graph with girth g and r max 1 r r 1 analogue results involving n 1 g and 1 g n 1 g to characterize vertex connectivity of graphs with fixed girth and clique number are also presented former results in liu et al 2013 22 liu et al 2019 20 hong et al 2019 15 liu et al 2019 21 and abiad et al 2018 1 are improved or extended

eigenvalue problems of ginzburg landau operator in bounded domains in this paper we study the eigenvalue problems for the ginzburg landau operator with a large parameter in bounded domains in r2 under gauge invariant boundary conditions the estimates for the eigenvalues are obtained and the asymptotic behavior of the associated eigenfunctions is discussed these results play a key role in estimating the critical magnetic field in the mathematical theory of superconductivity