energy of line graphs the energy of a graph is equal to the sum of the absolute values of its eigenvalues the energy of a matrix is equal to the sum of its singular values we establish relations between the energy of the line graph of a graph g and the energies associated with the laplacian and signless laplacian matrices of g

maximizing the least signless laplacian eigenvalue of unicyclic graphs abstract in this paper a graph with the maximum least signless laplacian eigenvalue among all connected unicyclic graphs with fixed order is determined

the least eigenvalue of graphs whose complements have only two pendent vertices abstract let g be a simple graph and a g be the adjacency matrix of g the eigenvalues of a g are referred to as the eigenvalues of g in this paper we characterize the graphs with the minimal least eigenvalue among all graphs whose complements are connected and have only two pendent vertices

escape rates in hamiltonian systems particles escape from the vicinity of invariant tori is macroscopically modeled by a nonuniform diffusive process the space dependence of transport coefficients is fixed by using perturbation theory scalings this leads to universal predictions for the escape rates that are then observed in numerical simulations

the smallest eigenvalue of the signless laplacian recently the signless laplacian matrix of graphs has been intensively investigated while there are many results about the largest eigenvalue of the signless laplacian the properties of its smallest eigenvalue are less well studied the present paper surveys the known results and presents some new ones about the smallest eigenvalue of the signless laplacian

on the numerical resolution of the generalized airfoil equation with possio kernel in this paper we consider the so called generalized airfoil singular integral equation with a smooth input function and present ad hoc quadrature rules to compute efficiently the elements of collocation and galerkin matrices

nonconvergence of formal integrals ii improved estimates for the optimal order of truncation we investigate the asymptotic properties of formal integral series in the neighbourhood of an elliptic equilibrium in nonlinear 2 dof hamiltonian systems in particular we study the dependence of the optimal order of truncation nopt on the distance from the elliptic equilibrium by numerical and analytical means the function nopt determines the region of nekhoroshev stability of the orbits and the time of practical stability we find that the function nopt decreases by abrupt steps the decrease is roughly approximated with an average power law nopt o a with a 1 we find an analytical explanation of this behaviour by investigating the accumulation of small divisors in both the normal form algorithm via lie series and in the direct construction of first integrals precisely we find that the series exhibit an apparent radius of convergence that tends to zero by abrupt steps as the order of the series tends to infinity our results agree with those obtained by servizi g et al 1983 phys lett a 95 11 for a conservative map of the plane moreover our analytical considerations allow us to explain the results of our previous paper contopoulos g et al 2003 j phys a math gen 36 8639 including in particular the different behaviour observed for low order and higher order resonances

signed bicyclic graphs minimizing the least laplacian eigenvalue abstract a signed graph is a pair g where g v g e g is a graph and e g 1 1 is the sign function on the edges of g for a signed graph we consider the laplacian matrix defined as l d g a where d g is the matrix of vertices degrees of g and a is the signed adjacency matrix the least laplacian eigenvalue is zero if and only if the signed graph is balanced i e all cycles contain an even number of negative edges here we show that among the unbalanced bicyclic signed graphs of given order n 5 the least laplacian eigenvalue is minimal for signed graphs consisting of two triangles only one of which is unbalanced connected by a path we also identify the signed graphs minimizing the least eigenvalue among those whose unbalanced bicyclic base is a theta graph

a note on the positive semidefiniteness of aα g abstract let g be a graph with adjacency matrix a g and let d g be the diagonal matrix of the degrees of g for every real 0 1 write a g for the matrix a g d g 1 a g let 0 g be the smallest for which a g is positive semidefinite it is known that 0 g 1 2 the main results of this paper are 1 if g is d regular then 0 min a g d min a g where min a g is the smallest eigenvalue of a g 2 g contains a bipartite component if and only if 0 g 1 2 3 if g is r colorable then 0 g 1 r

graph laplacians nodal domains and hyperplane arrangements eigenvectors of the laplacian of a graph g have received increasing attention in the recent past here we investigate their so called nodal domains i e the connected components of the maximal induced subgraphs of g on which an eigenvector psi does not change sign an analogue of courant s nodal domain theorem provides upper bounds on the number of nodal domains depending on the location of psi in the spectrum this bound however is not sharp in general in this contribution we consider the problem of computing minimal and maximal numbers of nodal domains for a particular graph the class of boolean hypercubes is discussed in detail we find that despite the simplicity of this graph class for which complete spectral information is available the computations are still non trivial nevertheless we obtained some new results and a number of conjectures author s abstract

on the laplacian and signless laplacian spectrum of a graph with k pairwise co neighbor vertices abstract consider the laplacian and signless laplacian spectrum of a graph g of order n with k pairwise co neighbor vertices we prove that the number of shared neighbors is a laplacian and a signless laplacian eigenvalue of g with multiplicity at least k 1 additionally considering a connected graph g k with a vertex set defined by the k pairwise co neighbor vertices of g the laplacian spectrum of g k obtained from g adding the edges of g k includes l for each nonzero laplacian eigenvalue of g k the laplacian spectrum of g overlaps the laplacian spectrum of g k in at least n k 1 places

the aα spectral radius characterization of some digraphs abstract let d be the a spectral radius of digraph d and let g n r be the set of digraphs with order n and dichromatic number r in this paper we characterize the digraph which has the maximal a spectral radius in g n r moreover we also determine the unique digraph which attains the maximum resp minimum a spectral radius among all strongly connected bicyclic digraphs

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

signless laplacians of finite graphs we survey properties of spectra of signless laplacians of graphs and discuss possibilities for developing a spectral theory of graphs based on this matrix for regular graphs the whole existing theory of spectra of the adjacency matrix and of the laplacian matrix transfers directly to the signless laplacian and so we consider arbitrary graphs with special emphasis on the non regular case the results which we survey old and new are of two types a results obtained by applying to the signless laplacian the same reasoning as for corresponding results concerning the adjacency matrix b results obtained indirectly via line graphs among other things we present eigenvalue bounds for several graph invariants an interpretation of the coefficients of the characteristic polynomial a theorem on powers of the signless laplacian and some remarks on star complements

on the aα spectral radius of a graph abstract let g be a graph with adjacency matrix a g and let d g be the diagonal matrix of the degrees of g for any real 0 1 nikiforov 3 defined the matrix a g as a g d g 1 a g the largest eigenvalue of a g is called the a spectral radius of g in this paper we give three edge graft transformations on a spectral radius as applications we determine the unique graph with maximum a spectral radius among all connected graphs with diameter d and determine the unique graph with minimum a spectral radius among all connected graphs with given clique number in addition some bounds on the a spectral radius are obtained

robust point pattern matching based on spectral context finding correspondences between two related feature point sets is a basic task in computer vision and pattern recognition in this paper we present a novel method for point pattern matching via spectral graph analysis in particular we aim to render the spectral matching algorithm more robust for positional jitter and outlier a local structural descriptor namely the spectral context is proposed to describe the attribute domain of point sets which is fundamentally different from the previous methods furthermore the approximate distance order is defined and employed as the metric for geometric consistency of neighboring points in this work by combining these two novel ingredients we formulate feature point set matching as an optimization problem with one to one constraints the correspondences are then obtained by maximizing the given objective function via the technique of probabilistic relaxation comparative experiments conducted on both synthetic and real data demonstrate the effectiveness of the proposed method especially in the presence of positional jitter and outliers a novel spectral method is proposed for the problem of point pattern matching we propose a local structural descriptor via spectral graph analysis we define a metric of geometric consistency by using the quantity of order

signless laplacian energy of a graph and energy of a line graph abstract for a simple graph g of order n size m and with signless laplacian eigenvalues q 1 q 2 q n the signless laplacian energy q e g is defined as q e g i 1 n q i d where d 2 m n is the average vertex degree of g we obtain the lower bounds for q e g in terms of first zagreb index m 1 g maximum degree d 1 second maximum degree d 2 minimum degree d n and second minimum degree d n 1 as a consequence of these bounds we obtain several bounds for the energy e l g of the line graph l g of graph g in terms of various graph parameters like m 1 g the clique number n m etc which improve some recently known bounds

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

commutativity and spectra of hermitian matrices abstract if two hermitian matrices commute then the eigenvalues of their sum are just the sums of the eigenvalues of the two matrices in a suitable order examples show that the converse is not true in general in this paper partial converses are obtained the technique involves a characterization of the equality cases for weyl s inequalities moreover a new proof on the commutativity of two hermitian matrices with property l and analogous results for the product of two positive definite hermitian matrices are included

on the relationship between the bruno function and the breakdown invariant tori we study the ratio epsilon c omega exp eta b omega where epsilon c omega is the breakdown threshold function for invariant tori eta is a parameter and b omega is the bruno function which is purely arithmetic i e it only depends by the continued fractions expansion of number omega we consider the standard map as a model and we focus our analysis on the exponential decay of the chaotic regions close to an invariant torus with diophantine rotation frequency our numerical experiments together with some heuristic consideration show that epsilon c omega exp eta b omega is not a continuous function on diophantine numbers omega for all values of eta

galerkin methods with splines for singular integral equations over 0 1 in this paper a convergence analysis of galerkin methods with splines for strongly elliptic singular integral equations over the interval 0 1 is given as trial functions we utilize smoothest polynomial splines on arbitrary meshes and continuous splines on special nonuniform partitions multiplied by a weight function using inequalities of garding type for singular integral operators in weightedl 2 spaces and the complete asymptotics of solutions at the endpoints we provide error estimates in certain sobolev norms

stieltjes polynomials and related quadrature rules in this paper we present a survey of theoretical results some of them new and numerical evidence of others concerning stieltjes polynomials kronrod schemes and their generalizations applied to integrals with a classical nonnegative weight function the use of these schemes in automatic integration routines is briefly outlined

the least eigenvalue of signless laplacian of graphs under perturbation abstract we investigate how the least eigenvalue of the signless laplacian of a graph changes by relocating a bipartite branch from one vertex to another vertex and minimize the least eigenvalue of the signless laplacian among the class of connected graphs with fixed order which contains a given non bipartite graph as an induced subgraph

product integration for the linear transport equation in slab geometry in this paper we present a product quadrature rule for the discretization of the well known linear transport equation in slab geometry in particular we give an algorithm for constructing the weights of the rule and prove that the order of convergence iso n 3 0 small as we like numerical examples are given and our formula is also compared with product simpson rules finally we examine a nystrom method based on our quadrature

properties of interpolatory product integration rules this paper is concerned with the approximate evaluation of int 1 1 k x f x d x where k is lebesgue integrable and f is at least riemann integrable and preferably smooth the integral is approximated by a rule of the form sum nolimits i 1 n w ni k f x ni where the points x ni are chosen in some prescribed way and the weights w ni k are such that the rule is exact if f is any polynomial of degree n for suitable choices of the points and suitable functions k the rule is shown to converge to the exact integral and the companion rule sum w ni k f x ni to int k x f x dx for all riemann integrable functions f the companion property ensures for example that the weights have the asymptotic positivity property if k is positive error bounds are given that guarantee rapid convergence of the rule if f is smooth finally by establishing a connection with mean convergence of lagrangian interpolation some negative theorems are proved in pa

on the role of high order resonances in normal forms and in separatrix splitting we discuss the role of high order resonances in the construction of normal forms for e close to integrable hamiltonian systems by heuristic considerations based on standard estimates we show that the remainder of normal forms is dominated by the terms corresponding to the main high order resonances and we provide a general argument to show that the size of such leading terms is exponentially small in 1e we apply this method to the problem of estimating the splitting of separatrices in resonant perturbed systems

a numerical study of arnold diffusion in a priori unstable systems this paper concerns the problem of the numerical detection of arnold diffusion in a priori unstable systems specifically we introduce a new definition of arnold diffusion which is adapted to the numerical investigation of the problem and is based on the numerical computation of the stable and unstable manifolds of the system examples of this arnold diffusion are provided in a model system in this model we also find that arnold diffusion behaves as an approximate markovian process thus it becomes possible to compute diffusion coefficients the values of the diffusion coefficients satisfy the scaling d epsilon simeq epsilon 2 we also find that this law is correlated to the validity of the melnikov approximation in fact the d epsilon simeq epsilon 2 law is valid up to the same critical value of e for which the error terms of melnikov approximations have a sharp increment

bipartiteness and the least eigenvalue of signless laplacian of graphs let g be a simple graph and let b g the least eigenvalue of the signless laplacian of the graph g in this paper we focus on the relations between the least eigenvalue and some parameters reflecting the graph bipartiteness we introduce two parameters the vertex bipartiteness b g and the edge bipartiteness b g and show that b g b g b g we also define another parameter g involved with a cut set and prove that b g g g 2 g 2 where g is the maximum degree of the graph g the above two inequalities are very similar in form to those given by fiedler and mohar respectively with respect to the algebraic connectivity of laplacian of graphs which is used to characterize the connectedness of graphs

the least signless laplacian eigenvalue of non bipartite graphs with given stability number abstract in this paper we investigate the least eigenvalue of signless laplacian matrix of a connected graph and determine the graphs which have the minimum least signless laplacian eigenvalue among all nonbipartite graphs with given stability number n 1 2 or covering number 2 n 1 2 respectively

chirikov and nekhoroshev diffusion estimates bridging the two sides of the river abstract we present theoretical and numerical results pointing towards a strong connection between the estimates for the diffusion rate along simple resonances in multidimensional nonlinear hamiltonian systems that can be obtained using the heuristic theory of chirikov and a more formal one due to nekhoroshev we show that despite a wide spread impression the two theories are complementary rather than antagonist indeed although chirikov s 1979 review has thousands of citations almost all of them refer to topics such as the resonance overlap criterion fast diffusion the standard or whisker map and not to the constructive theory providing a formula to measure diffusion along a single resonance however as will be demonstrated explicitly below chirikov s formula provides values of the diffusion coefficient which are quite well comparable to the numerically computed ones provided that it is implemented on the so called optimal normal form derived as in the analytic part of nekhoroshev s theorem on the other hand chirikov s formula yields unrealistic values of the diffusion coefficient in particular for very small values of the perturbation when used in the original hamiltonian instead of the optimal normal form in the present paper we take advantage of this complementarity in order to obtain accurate theoretical predictions for the local value of the diffusion coefficient along a resonance in a specific 3dof nearly integrable hamiltonian system besides we compute numerically the diffusion coefficient and a full comparison of all estimates is made for ten values of the perturbation parameter showing a very satisfactory agreement

bounds of signless laplacian spectrum of graphs based on the k domination number abstract let g v g e g be a graph a k dominating set in g is a subset x of v g such that each element of v g x is adjacent to at least k vertices of x the least cardinality of a k dominating set is the k domination number in this paper we will give some bounds of the signless laplacian spectrum of g involving the k domination number

bounds and conjectures for the signless laplacian index of graphs abstract using the autographix system we obtain conjectures of the form l n q 1 i g u n where q 1 denotes the signless laplacian index of graph g is one the four operations i g is another invariant chosen among minimum average and maximum degree average distance diameter radius girth proximity remoteness vertex edge and algebraic connectivities independence number domination number clique number chromatic number and matching number randic index l n and u n are best possible lower and upper bounds function of the order n of g algebraic conjectures are obtained in 120 cases out of 152 and structural conjectures in 12 of the remaining cases these conjectures are known immediate or proved in this paper except for 17 of them which remain open

on the aα spectra of graphs abstract let g be a graph with adjacency matrix a g and let d g be the diagonal matrix of the degrees of g for any real 0 1 nikiforov 8 defined the matrix a g as a g d g 1 a g in this paper we give some results on the eigenvalues of a g for 1 2 in particular we characterize the graphs with k a g n 1 for 2 k n moreover we show that n a g 2 1 if g contains no isolated vertices

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

the domination number and the least q eigenvalue abstract in this paper for a nonbipartite graph with both order n and domination number we show that it contains a nonbipartite unicyclic spanning subgraph with the same domination number we also present some results about the changing of the domination number when the structure of a graph is changed by investigating the relation between the domination number and the least q eigenvalue of a graph we minimize the least q eigenvalue among all the nonbipartite graphs with both order n 4 and given domination number n 1 3

on the least eigenvalue of aα matrix of graphs abstract let g be a graph of order n with m edges and chromatic number let a g be the adjacency matrix and d g be the diagonal matrix of vertex degrees of g nikiforov defined the matrix a g as a g d g 1 a g where 0 1 then a 1 2 g 1 2 d g a g 1 2 q g where q g is the signless laplacian matrix of g let q n g and n a be the least eigenvalue of q g and a g respectively lima et al 2011 8 proposed some open problems on q n g two of which are as follows 1 to characterize the graphs for which q n g 2 m n 1 2 to characterize the graphs for which q n g 1 1 1 2 m n in this paper we present an upper bound on n a in terms of n m and 1 2 1 and characterize the extremal graphs as an application we completely solve problem 1 moreover we examine the more generalized result of problem 2 on a g when 1 we obtain some necessary conditions for n a 1 2 m 1 n and as a corollary for the equality in problem 2

the classical collocation method for singular integral equations first of all a method of collocation which we call classical collocation is described for the approximate solution of complete singular integral equations with cauchy kernel taken over the arc 1 1 secondly we demonstrate that under reasonable conditions the approximate solutions converge to the solution of the original equation

signed graphs with extremal least laplacian eigenvalue abstract a signed graph is a pair g where g v g e g is a graph and e g is the corresponding sign function for a signed graph we consider the laplacian matrix defined as l d g a where d g is the matrix of vertex degrees of g and a is the signed adjacency matrix it is well known that is balanced that is each cycle contains an even number of negative edges if and only if the least laplacian eigenvalue n 0 therefore if is not balanced then n 0 we show here that among unbalanced connected signed graphs of given order the least eigenvalue is minimal for an unbalanced triangle with a hanging path while the least eigenvalue is maximal for the complete graph with the all negative sign function

a note on the aα spectral radius of graphs abstract let g be a graph with adjacency matrix a g and let d g be the diagonal matrix of the degrees of g for any real 0 1 nikiforov 2017 7 defined the matrix a g as a g d g 1 a g let u and v be two vertices of a connected graph g suppose that u and v are connected by a path w 0 v w 1 w s 1 w s u where d w i 2 for 1 i s 1 let g p s q u v be the graph obtained by attaching the paths p p to u and p q to v let s 0 1 nikiforov and rojo 2018 9 conjectured that g p s q u v g p 1 s q 1 u v if p q 2 in this paper we confirm the conjecture as applications firstly the extremal graph with maximal a spectral radius with fixed order and cut vertices is characterized secondly we characterize the extremal tree which attains the maximal a spectral radius with fixed order and matching number these results generalize some known results

an asymptotic expression for the splitting of separatrices of the rapidly forced pendulum the measure of the splitting of the separatrices of the rapidly forced pendulum ddot x sin x mu sin frac t varepsilon is considered as a model problem that has been studied by different authors here e are small parameters e 0 but otherwise independent the following formula for the angle between separatrices is established alpha frac pi 2 varepsilon frac mu cosh frac pi 2 varepsilon left 1 o mu varepsilon 2 right

further convergence results for the weighted galerkin method of numerical solution of cauchy type singular integral equations the convergence of the weighted lalerkin method based on chcbvshen or jacobi polvnoinials for the direct numerical solution of one dimensional real cauchv tvpe singular integral equations of the first and of the second kind on a finite interval is proved under sufficientlv weak continuitv assumptions for the kernels and the right side functions of thc integral equations

numerical solution of the generalized airfoil equation for an airfoil with a flap in this paper we are concerned with the generalized airfoil equation which governs the pressure across an airfoil oscillating in a wind tunnel in particular we examine the case of an airfoil which contains a flap so that the input function has a simple jump at a point x0 1 x0 1 this generates a logarithmic singularity in the solution u at the same point x0 and considerably complicates the problem of the numerical approximation of u by introducing a nonlinear transformation of the independent variable in the integral equation we improve the behavior of the unknown function u through incorporation of the jacobian of the transformation the price is that the kernel of the new integral equation becomes more complicated nevertheless we are able to show that a galerkin method based on high order polynomials is both stable and convergent we also present some promising numerical results for the corresponding collocation method for which we have not succeeded in proving stability and convergence although those results seem to suggest that it is indeed stable and has a rate of convergence similar to that we have derived for the galerkin method

a sharp lower bound for the least eigenvalue of the signless laplacian of a non bipartite graph we prove that the minimum value of the least eigenvalue of the signless laplacian of a connected non bipartite graph with a prescribed number of vertices is attained solely in the unicyclic graph obtained from a triangle by attaching a path at one of its endvertices

maximizing signless laplacian or adjacency spectral radius of graphs subject to fixed connectivity in this paper we characterize the graphs with maximum signless laplacian or adjacency spectral radius among all graphs with fixed order and given vertex or edge connectivity we also discuss the minimum signless laplacian or adjacency spectral radius of graphs subject to fixed connectivity consequently we give an upper bound of signless laplacian or adjacency spectral radius of graphs in terms of connectivity in addition we confirm a conjecture of aouchiche and hansen involving adjacency spectral radius and connectivity

the least signless laplacian eignvalue of the complements of unicyclic graphs abstract let s n c be the set of all connected graph each of which is a complement of an n vertex unicyclic graph li and wang 2012 determined the graphs with the least signless laplacian eignvalue among all the graphs of the complements of n vertex trees in this paper as a continuance of it the unique graph among s n c which minimizes the least signless laplacian eigenvalue is identified

the aα spread of a graph abstract let a g and d g be the adjacency matrix and the degree diagonal matrix of a graph g respectively for any real number 0 1 nikiforov defined the a matrix of a graph g as a g d g 1 a g the a spread of a graph is the difference between the largest eigenvalue and the smallest eigenvalue of the a matrix of the graph in this paper we obtain some new lower bounds on a spread and characterize the extremal graphs for certain cases which extend the results of a spread and q spread moreover some graph operations on a spread are presented

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

on the extremal values of the second largest q eigenvalue we study extremal graphs for the extremal values of the second largest q eigenvalue of a connected graph we first characterize all simple connected graphs with second largest signless laplacian eigenvalue at most 3 the second part of the present paper is devoted to the study of the graphs that maximize the second largest q eigenvalue we construct families of such graphs and prove that some of theses families are minimal for the fact that they maximize the second largest signless laplacian eigenvalue

on the aα spectra of trees abstract let g be a graph with adjacency matrix a g and let d g be the diagonal matrix of the degrees of g for every real 0 1 define the matrix a g as a g d g 1 a g this paper gives several results about the a matrices of trees in particular it is shown that if t is a tree of maximal degree then the spectral radius of a t satisfies the tight inequality a t 2 1 1 which implies previous bounds of godsil lovasz and stevanovic the proof is deduced from some new results about the a matrices of bethe trees and generalized bethe trees in addition several bounds on the spectral radius of a of general graphs are proved implying tight bounds for paths and bethe trees

lower bounds for the spread of a hermitian matrix abstract some simple lower bounds for the spread of a hermitian matrix are derived these bounds are explicit functions of the entries of the matrix one of which is sharper than a recent result due to e r barnes and a j hoffman comparisons are made with several known results

on the least q eigenvalue of a non bipartite hamiltonian graph abstract in this paper we study the least q eigenvalue of a non bipartite hamiltonian graph and determine all non bipartite hamiltonian graphs whose least q eigenvalue attains the minimum among all non bipartite hamiltonian graphs on n vertices

characterizations and lower bounds for the spread of a normal matrix abstract the spread of an n n matrix a with eigenvalues 1 n is defined by spr a max j k j k we prove that if a is normal then spr a max x ax y ay x y c n x y 1 max x ay y ax x y c n x y 1 re x y 0 max x ay y ax x y c n x y 1 x y 0 max spr z a z a 2 z c z 1 2 max x a 2 x x ax 2 x a ax x ax 2 1 2 x c n x 1 we also present several lower bounds for spr a given by these characterizations

bounds for the largest and the smallest aα eigenvalues of a graph in terms of vertex degrees abstract let g be a graph with adjacency matrix a g and with d g the diagonal matrix of its vertex degrees nikiforov defined the matrix a g with 0 1 as a g d g 1 a g the largest and the smallest eigenvalues of a g are respectively denoted by g and n g in this paper we present a tight upper bound for g in terms of the vertex degrees of g for 1 for any graph g g max u w d u d w 2 d u d w 2 4 1 2 d u d w 2 if g is connected for 0 1 the equality holds if and only if g is regular or bipartite semi regular as an application we solve the problem raised by nikiforov in 15 for the smallest eigenvalue n g of a bipartite graph g of order n with no isolated vertices for 0 1 then n g min u w d u d w 2 d u d w 2 4 1 2 d u d w 2 if g is connected and 1 2 the equality holds if and only if g is regular or semi regular

convergence and stability of a collocation method for the generalized airfoil equation we establish the mean square convergence and numerical stability of a collocation method for solving a class of cauchy singular integral equations the method uses the jacobi polynomials p 1 2 1 2 n as basis elements and the zeros of the polynomial p 1 2 1 2 n 1 as collocation points uniform convergence is shown to hold for the first iterate convergence is proved via vainikko s embedding method and stability is demonstrated by generalizing the approach of atkinson for integral equations of the second kind

proof of conjecture involving the second largest signless laplacian eigenvalue and the index of graphs abstract let g v e be a simple graph denote by d g the diagonal matrix of its vertex degrees and by a g its adjacency matrix then the signless laplacian matrix of g is q g d g a g in 5 cvetkovic et al have given the following conjecture involving the second largest signless laplacian eigenvalue q 2 and the index 1 of graph g see also aouchiche and hansen 1 1 n 1 q 2 1 n 2 2 n 4 with equality if and only if g is the star k 1 n 1 for the lower bound and if and only if g is the complete bipartite graph k n 2 2 for the upper bound in this paper we prove the lower bound and characterize the extremal graphs

on the least distance eigenvalue of a graph abstract denote by d g d i j n n the distance matrix of a connected graph g with n vertices where d i j is equal to the distance between vertices v i and v j in g the least eigenvalue of d g is called the least distance eigenvalue of g denoted by n in this paper we determine all the graphs with n 2 383 0

spectral characterization of some weighted rooted graphs with cliques abstract the level of a vertex in a rooted graph is one more than its distance from the root vertex a generalized bethe tree is a rooted tree in which vertices at the same level have the same degree we characterize completely the eigenvalues of the laplacian signless laplacian and adjacency matrices of a weighted rooted graph g obtained from a weighted generalized bethe tree of k levels and weighted cliques in which 1 the edges connecting vertices at consecutive levels have the same weight 2 each set of children in one or more levels defines a weighted clique and 3 cliques at the same level are isomorphic these eigenvalues are the eigenvalues of symmetric tridiagonal matrices of order j j 1 j k moreover we give results on the multiplicity of the eigenvalues on the spectral radii and on the algebraic conectivity finally we apply the results to the unweighted case and some particular graphs are studied

the smallest signless laplacian spectral radius of graphs with a given clique number abstract the signless laplacian spectral radius of a graph g is the largest eigenvalue of its signless laplacian matrix in this paper the first four smallest values of the signless laplacian spectral radius among all connected graphs with maximum clique of size greater than or equal to 2 are obtained

which graphs are determined by their spectrum for almost all graphs the answer to the question in the title is still unknown here we survey the cases for which the answer is known not only the adjacency matrix but also other types of matrices such as the laplacian matrix are considered

new bounds for the signless laplacian spread abstract let g be an undirected simple graph the signless laplacian spread of g is defined as the maximum distance of pairs of its signless laplacian eigenvalues this paper establishes some new bounds both lower and upper for the signless laplacian spread several of these bounds depend on invariant parameters of the graph we also use a minmax principle to find several lower bounds for this spectral invariant

the least q eigenvalue with fixed domination number abstract denote by lg l the lollipop graph obtained by attaching a pendant path p v g v g 1 v g l l 1 to a cycle c v 1 v 2 v g v 1 g 3 a f g l g r a p h of order n g 1 is defined to be the graph obtained by attaching n g l pendent vertices to some of the nonpendant vertices of lg l in which each vertex other than v g l 1 is attached at most one pendant vertex a f g l graph is a f g l g r a p h in which vg is attached with pendant vertex denote by qmin the least q e i g e n v a l u e of a graph in this paper we proceed on considering the domination number the least q eigenvalue of a graph as well as their relation further results obtained are as follows i some results about the changing of the domination number under the structural perturbation of a graph are represented ii among all nonbipartite unicyclic graphs of order n with both domination number and girth g g n 1 the minimum qmin attains at a f g l graph for some l iii among the nonbipartite graphs of order n and with given domination number which contain a f g l graph as a subgraph some lower bounds for qmin are represented iv among the nonbipartite graphs of order n and with given domination number n 2

eigenvalues of certain weighted graphs joined at their roots having cliques at some levels abstract a generalized bethe tree is a rooted tree in which vertices at the same level have the same degree for i 1 2 p let b i be a generalized bethe tree of k i levels and let i 1 2 k i 1 such that 1 the edges of b i connecting vertices at consecutive levels have the same weight and 2 for j i each set of children of b i at the level k i j 1 defines a clique in which the edges have weight u i j for i 1 2 p let g i be the graph obtained from b i and the cliques at the levels k i j 1 for all j i let g be the graph obtained from the graphs g i 1 i p joined at their respective roots we give a complete characterization of the eigenvalues including their multiplicities of the laplacian signless laplacian and adjacency matrices of the graph g finally we characterize the normalized laplacian eigenvalues when g is an unweighted graph

on the α index of graphs with pendent paths abstract let g be a graph with adjacency matrix a g and let d g be the diagonal matrix of the degrees of g for every real 0 1 write a g for the matrix a g d g 1 a g this paper presents some extremal results about the spectral radius g of a g that generalize previous results about 0 g and 1 2 g in particular write b p q r be the graph obtained from a complete graph k p by deleting an edge and attaching paths p q and p r to its ends it is shown that if 0 1 and g is a graph of order n and diameter at least k then g b n k 2 k 2 k 2 with equality holding if and only if g b n k 2 k 2 k 2 this result generalizes results of hansen and stevanovic 5 and liu and lu 7

bounds on the q spread of a graph the spread s m of an n n complex matrix m is s m maxij i j where the maximum is taken over all pairs of eigenvalues of m i 1 i n 9 and 11 based on this concept gregory et al 7 determined some bounds for the spread of the adjacency matrix a g of a simple graph g and made a conjecture regarding the graph on n vertices yielding the maximum value of the spread of the corresponding adjacency matrix the signless laplacian matrix of a graph g q g d g a g where d g is the diagonal matrix of degrees of g and a g is its adjacency matrix has been recently studied 4 5 the main goal of this paper is to determine some bounds on s q g we prove that for any graph on n 5 vertices 2 s q g 2n 4 and we characterize the equality cases in both bounds further we prove that for any connected graph g with n 5 vertices s q g 2n 4 we conjecture that for n 5 sq g 4n 2 20n 33 and that in this case the upper bound is attained if and only if g is a certain pathcomplete graph

on the multiplicity of α as an eigenvalue of aα g of graphs with pendant vertices abstract let g be a simple undirected graph let 0 1 let a g d g 1 a g where d g and a g are the diagonal matrix of the vertex degrees of g and the adjacency matrix of g respectively let p g 0 and q g be the number of pendant vertices and quasi pendant vertices of g respectively let m g be the multiplicity of as an eigenvalue of a g it is proved that m g p g q g with equality if each internal vertex is a quasi pendant vertex if there is at least one internal vertex which is not a quasi pendant vertex the equality m g p g q g m n is determined in which m n is the multiplicity of as an eigenvalue of the matrix n this matrix is obtained from a g taking the entries corresponding to the internal vertices which are non quasi pendant vertices these results are applied to search for the multiplicity of as an eigenvalue of a g when g is a path a caterpillar a circular caterpillar a generalized bethe tree or a bethe tree for the bethe tree case a simple formula for the nullity is given

lower bounds for the minimum eigenvalue of hadamard product of an m matrix and its inverse for the hadamard product a a 1 of an m matrix a and its inverse a 1 we give new lower bounds for the minimum eigenvalue of a a 1 these bounds are strong enough to prove the conjecture of fiedler and markham an inequality for the hadamard product of an m matrix and inverse m matrix linear algebra appl 101 1988 1 8

spectra of copies of a generalized bethe tree attached to any graph abstract a generalized bethe tree is a rooted unweighted tree in which vertices at the same level have the same degree let g be any connected graph let g b be the graph obtained from g by attaching a generalized bethe tree b by its root to each vertex of g we characterize completely the eigenvalues of the signless laplacian laplacian and adjacency matrices of the graph g b including results on the eigenvalue multiplicities finally for the laplacian and signless laplacian matrices we recall a procedure to compute a tight upper bound on the algebraic connectivity of g b as well as on the smallest eigenvalue of the signless laplacian matrix of g b whenever g is a non bipartite graph

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

on the sign patterns of the smallest signless laplacian eigenvector abstract let h be a connected bipartite graph whose signless laplacian matrix is q h suppose that the bipartition of h is s t and that x is the eigenvector of the smallest eigenvalue of q h it is well known that x is positive and constant on s and negative and constant on t the resilience of the sign pattern of x under addition of edges into the subgraph induced by either s or t is investigated and a number of cases in which the sign pattern of x persists are described

a lower bound on the least signless laplacian eigenvalue of a graph let g be a simple graph with n vertices and m edges and let q n g be the least signless laplacian eigenvalue of g recently guo chen and yu proved that if n ge 6 then begin aligned q n g ge frac 2m n 2 n 1 end aligned which confirms a conjecture proposed by de lima oliveira de abreu and nikiforov in this note we present a new proof for the above inequality where we also completely characterize the case when the equality holds