the spectrum and the signless laplacian spectrum of coronae abstract let g 1 g 2 be two simple connected graphs denote the corona and the edge corona of g 1 g 2 by g 1 g 2 and g 1 g 2 respectively in this paper we first introduce a new invariant the m coronal of a graph matrix m where the matrix m is associated with a graph in a prescribed way then it is used to compute the signless laplacian spectrum of g 1 g 2 and g 1 g 2 in terms of the signless laplacian spectrum of g 1 and g 2 in addition the spectrum of g 1 g 2 are also given in terms of the spectrum of g 1 and g 2 finally as an application of these results we construct many pairs of nonisomorphic signless laplacian cospectral graphs

criteria for quasi convexity and pseudo convexity relationships and comparisons a first order criterion for pseudo convexity and second order criteria for quasi convexity and pseudo convexity are given for twice differentiable functions on open convex sets the relationships between these second order criteria and other known criteria are also analysed finally the numbers of operations required to verify these criteria are calculated and compared

trees with minimal laplacian coefficients let g be a simple undirected graph with the characteristic polynomial of its laplacian matrix l g p g m email protected k 0 n 1 kc email protected n k it is well known that for trees the laplacian coefficient c n 2 is equal to the wiener index of g while c n 3 is equal to the modified hyper wiener index of the graph in this paper we characterize n vertex trees with given matching number m which simultaneously minimize all laplacian coefficients the extremal tree a n m is a spur obtained from the star graph s n m 1 with n m 1 vertices by attaching a pendant edge to each of certain m 1 non central vertices of s n m 1 in particular a n m minimizes the wiener index the modified hyper wiener index and the recently introduced incidence energy of trees defined as ie g email protected k 0 email protected k where m k are the eigenvalues of signless laplacian matrix q g d g a g we introduced a general r transformation which decreases all laplacian coefficients simultaneously in conclusion we illustrate on examples of wiener index and incidence energy that the opposite problem of simultaneously maximizing all laplacian coefficients has no solution

on state dependent implication in quantum mechanics this paper studies a generalization of the a priori implication e or f meaning of course ef e in the quantum logic p h of all projectors the new definition can be applied also to new pairs of events being based on the indistinguishability of some events when considered from the standpoint of a fixed state rho the mathematical theory of factor boolean algebras on which this extension is based is clarified

on isometric embeddings of graphs si g est un graphe connexe fini v e une facon standard de definir une distance d g sur g est de definir d g x y comme etant le nombre d aretes dans un chemin le plus court joignant x et y dans v on etudie la theorie des plongements isometriques de graphes dans des produits cartesiens d espaces metriques

roundness and metric type abstract we prove that if x is a banach space containing l n p uniformly in n and if y is a metric space with metric type q p then the inverse of any uniform homeomorphism t from x onto y cannot satisfy a lipschitz condition for large distances of order q p it follows that if y is a midpoint convex subset of a banach space z with type q larger than the type supremum of a banach space x then x and y cannot be uniformly homeomorphic in particular we prove the non existence of uniform homeomorphisms between certain non commutative l p spaces and midpoint convex subsets of another such space we also prove that if a banach space x has cotype infimum q larger than two then it has maximal generalized roundness zero and maximal roundness at most q as a consequence infinite dimensional c algebras are seen to have maximal generalized roundness zero and maximal roundness one

on the reduced signless laplacian spectrum of a degree maximal graph abstract for a simple graph g the signless laplacian of g is the matrix a g d g where a g is the adjacency matrix and d g is the diagonal matrix of vertex degrees of g the reduced signless laplacian of g is the matrix g b g where b g is the reduced adjacency matrix of g and g is the diagonal matrix whose diagonal entries are the common degrees for vertices belonging to the same neighborhood equivalence class of g a graph is said to be degree maximal if it is connected and its degree sequence is not majorized by the degree sequence of any other connected graph for a maximal graph we obtain a formula for the characteristic polynomial of its reduced signless laplacian and use the formula to derive a localization result for its reduced signless laplacian eigenvalues and to compare the signless laplacian spectral radii of two well known maximal graphs we also obtain a necessary condition for a maximal graph to have maximal signless laplacian spectral radius among all connected graphs with given numbers of vertices and edges

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

expander flows geometric embeddings and graph partitioning we give a o slog n approximation algorithm for the sparsest cut edge expansion balanced separator and graph conductance problems this improves the o log n approximation of leighton and rao 1988 we use a well known semidefinite relaxation with triangle inequality constraints central to our analysis is a geometric theorem about projections of point sets in rd whose proof makes essential use of a phenomenon called measure concentration we also describe an interesting and natural approximate certificate for a graph s expansion which involves embedding an n node expander in it with appropriate dilation and congestion we call this an expander flow

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

sharp bounds for the largest eigenvalue of the signless laplacian of a graph let g be a simple connected graph with n vertices and m edges denote the degree of vertex vi by d vi the matrix q g d g a g is called the signless laplacian of g where d g diag d v1 d v2 d vn and a g denote the diagonal matrix of vertex degrees and the adjacency matrix of g respectively let q1 g be the largest eigenvalue of q g in this paper we first present two sharp upper bounds for q1 g involving the maximum degree and the minimum degree of the vertices of g and give a new proving method on another sharp upper bound for q1 g then we present three sharp lower bounds for q1 g involving the maximum degree and the minimum degree of the vertices of g moreover we determine all extremal graphs which attain these sharp bounds

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

integral complete r partite graphs a graph is called integral if all the eigenvalues of its adjacency matrix are integers in this paper we give a useful sufficient and necessary condition for complete r partite graphs to be integral from which we can construct infinite many new classes of such integral graphs it is proved that the problem of finding such integral graphs is equivalent to the problem of solving some diophantine equations the discovery of these integral complete r partite graphs is a new contribution to the search of such integral graphs finally we propose several basic open problems for further study

the coarse baum connes conjecture for spaces which admit a uniform embedding into hilbert space corollary 1 2 let be a finitely generated group if as a metric space with a word length metric admits a uniform embedding into hilbert space and its classifying space b has the homotopy type of a finite cw complex then the strong novikov conjecture holds for i e the index map from k b to k c r is injective corollary 1 2 follows from theorem 1 1 and the descent principle 23 by index theory the strong novikov conjecture implies the novikov conjecture on the homotopy invariance of higher signatures cf 8 for an excellent

remarks on q integral complete multipartite graphs a graph is q integral if the spectrum of its signless laplacian matrix consists entirely of integers in their study of q integral complete multipartite graphs zhao et al q integral complete r partite graphs linear algebra appl 438 2013 1067 1077 posed two questions on the existence of such graphs we resolve these questions and present some further results characterizing particular classes of q integral complete multipartite graphs

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

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

starlike trees whose maximum degree exceed 4 are determined by their q spectra abstract let a g and d g be the adjacency matrix and the degree matrix of a graph g respectively the matrix d g a g is called the signless laplacian matrix of g the spectrum of the matrix d g a g is called the q spectrum of g a graph is said to be determined by its q spectrum if there is no other non isomorphic graph with the same q spectrum in this paper we prove that all starlike trees whose maximum degree exceed 4 are determined by their q spectra

on some forests determined by their laplacian or signless laplacian spectrum we consider the class of graphs whose each component is either a proper subgraph of some smith graphs or belongs to a precized subset of smith graphs we classify the graphs from the considered class into those which are determined or not determined by laplacian or signless laplacian spectrum

estimating the gap of finite metric spaces of strict p negative type abstract let x d be a finite metric space this paper first discusses the spectrum of the p distance matrix of a finite metric space of p negative type and then gives upper and lower bounds for the so called gap x p of a finite metric space of strict p negative type furthermore estimations for x p under a certain gluing construction for finite metric spaces are given and finally be applied to finite ultrametric spaces

new extremal problems for the riemannian recognition program via alexandrov geometry in its most general form the recognition problem in riemannian geometry asks for the identification of an unknown riemannian manifold via measurements of metric invariants on the manifold optimally one wants to recognize a manifold having made as few measurements as possible many results in riemannian geometry including pinching theorems can be viewed this way here we are only interested in measurements that assign real numbers to each complete riemannian manifold typical examples of such invariants are diameter volume curvature bounds etc when viewing one or several invariants i ii ii of this type as a map on a suitable class l of riemannian manifolds the following problems pose themselves

on distance matrices and laplacians we consider distance matrices of certain graphs and of points chosen in a rectangular grid formulae for the inverse and the determinant of the distance matrix of a weighted tree are obtained results concerning the inertia and the determinant of the distance matrix of an unweighted unicyclic graph are proved if d is the distance matrix of a tree then we obtain certain results for a perturbation of d 1 as an example it is shown that if l is the laplacian matrix of an arbitrary connected graph then d 1 l 1 is an entrywise positive matrix we consider the distance matrix of a subset of a rectangular grid of points in the plane if we choose m k 1 points not containing a closed path in an m k grid then a formula for the determinant of the distance matrix of such points is obtained

signless laplacian spectral radii of graphs with given chromatic number abstract let g be a simple graph with vertices v 1 v 2 v n of degrees d 1 d 2 d n respectively let a be the 0 1 adjacency matrix of g and d be the diagonal matrix diag d 1 d 2 d n q g d a is called the signless laplacian of g the largest eigenvalue of q g is called the signless laplacian spectral radius or q spectral radius of g denote by g the chromatic number for a graph g in this paper for graphs with order n the extremal graphs with both the given chromatic number and the maximal q spectral radius are characterized the extremal graphs with both the given chromatic number 4 5 6 7 and the minimal q spectral radius are characterized as well

trees and ultrametric spaces a categorical equivalence abstract there is a well known correspondence between infinite trees and ultrametric spaces that comes from considering the end space of the tree the correspondence is interpreted here as an equivalence between two categories one of which encodes the geometry of trees at infinity and the other encodes the micro geometry of complete ultrametric spaces

on products and line graphs of signed graphs their eigenvalues and energy abstract in this article we examine the adjacency and laplacian matrices and their eigenvalues and energies of the general product non complete extended p sum or neps of signed graphs we express the adjacency matrix of the product in terms of the kronecker matrix product and the eigenvalues and energy of the product in terms of those of the factor graphs for the cartesian product we characterize balance and compute expressions for the laplacian eigenvalues and laplacian energy we give exact results for those signed planar cylindrical and toroidal grids which are cartesian products of signed paths and cycles we also treat the eigenvalues and energy of the line graphs of signed graphs and the laplacian eigenvalues and laplacian energy in the regular case with application to the line graphs of signed grids that are cartesian products and to the line graphs of all positive and all negative complete graphs

on the laplacian coefficients of signed graphs abstract let g be a signed graph where g is its underlying graph and its sign function defined on edges of g a signed graph the subgraph of is its signed tu subgraph if the signed graph induced by the vertices of consists of trees and or unbalanced unicyclic signed graphs let l d g a be the laplacian of in this paper we express the coefficient of the laplacian characteristic polynomial of based on the signed tu subgraphs of and establish the relation between the laplacian characteristic polynomial of a signed graph with adjacency characteristic polynomials of its signed line graph and signed subdivision graph as an application we identify the signed unicyclic graphs having extremal coefficients of the laplacian characteristic polynomial

enhanced negative type for finite metric trees abstract a finite metric tree is a finite connected graph that has no cycles endowed with an edge weighted path metric finite metric trees are known to have strict 1 negative type in this paper we introduce a new family of inequalities 1 that encode the best possible quantification of the strictness of the non trivial 1 negative type inequalities for finite metric trees these inequalities are sufficiently strong to imply that any given finite metric tree t d must have strict p negative type for all p in an open interval 1 1 where 0 may be chosen so as to depend only upon the unordered distribution of edge weights that determine the path metric d on t in particular if the edges of the tree are not weighted then it follows that depends only upon the number of vertices in the tree we also give an example of an infinite metric tree that has strict 1 negative type but does not have p negative type for any p 1 this shows that the maximal p negative type of a metric space can be strict

the deformed consensus protocol this paper studies a generalization of the standard continuous time consensus protocol obtained by replacing the laplacian matrix of the communication graph with the so called deformed laplacian the deformed laplacian is a second degree matrix polynomial in the real variable s which reduces to the standard laplacian for s equal to unity the stability properties of the ensuing deformed consensus protocol are studied in terms of parameter s for some special families of undirected and directed graphs and for arbitrary graph topologies by leveraging the spectral theory of quadratic eigenvalue problems examples and simulation results are provided to illustrate our theoretical findings

on the gap of finite metric spaces of p negative type let x d be a metric space of p negative type recently i doust and a weston introduced a quantification of the p negative type property the so called gap of x this paper gives some formulas for the gap of a finite metric space of strict p negative type and applies them to evaluate for some concrete finite metric spaces

polygonal equalities and virtual degeneracy in lp spaces abstract suppose 0 p 2 and that is a measure space for which l p is at least two dimensional the central results of this paper provide a complete description of the subsets of l p that have strict p negative type in order to do this we study non trivial p polygonal equalities in l p these are equalities that can after appropriate rearrangement and simplification be expressed in the form j i 1 n j i z j z i p p 0 where z 1 z n is a subset of l p and 1 n are non zero real numbers that sum to zero we provide a complete classification of the non trivial p polygonal equalities in l p the cases p 2 and p 2 are substantially different and are treated separately the case p 1 generalizes an elegant result of elsner han koltracht neumann and zippin another reason for studying non trivial p polygonal equalities in l p is due to the fact that they preclude the existence of certain types of isometry for example our techniques show that if x d is a metric space that has strict q negative type for some q p then 1 x d is not isometric to any linear subspace w of l p that contains a pair of disjointly supported non zero vectors and 2 x d is not isometric to any subset of l p that has non empty interior furthermore in the case p 2 it also follows that x d is not isometric to any affinely dependent subset of l 2 more generally we show that if y is a metric space whose generalized roundness is finite and if x d is a metric space that has strict q negative type for some q then x d is not isometric to any metric subspace of y that admits a non trivial p 1 polygonal equality for some p 1 q it is notable in all of these statements that the metric space x d can for instance be any ultrametric space as a result we obtain new insights into sophisticated embedding theorems of lemin and shkarin we conclude the paper by constructing some pathological infinite dimensional linear subspaces of l p that do not have strict p negative type

characterization stability and convergence of hierarchical clustering methods we study hierarchical clustering schemes under an axiomatic view we show that within this framework one can prove a theorem analogous to one of kleinberg 2002 in which one obtains an existence and uniqueness theorem instead of a non existence result we explore further properties of this unique scheme stability and convergence are established we represent dendrograms as ultrametric spaces and use tools from metric geometry namely the gromov hausdorff distance to quantify the degree to which perturbations in the input metric space affect the result of hierarchical methods

circum euclidean distance matrices and faces abstract we study the structure of circum euclidean distance matrices those euclidean distance matrices generated by points lying on a hypersphere we show for example that such euclidean distance matrices are characterized as having constant row sums and they constitute the interior of the cone of all euclidean distance matrices also we provide a formula for computing the radius of a representing configuration in the smallest embedding dimension r and show that rk d r 1 finally we obtain a geometric characterization of the faces of this cone given a configuration of points and its euclidean distance matrix d any matrix in the minimal face containing d comes from a configuration that is a linear perturbation of the points that generate d

on the generalized roundness of finite metric spaces in this paper we show that the generalized roundness of a finite metric space can be bounded from below by a strictly positive constant that only depends upon the number of points in the space

asymptotic negative type properties of finite ultrametric spaces abstract negative type inequalities arise in the study of embedding properties of metric spaces but they often reduce to intractable combinatorial problems in this paper we study more quantitative versions of these inequalities involving the so called p negative type gap in particular we focus our attention on the class of finite ultrametric spaces which are important in areas such as phylogenetics and data mining let x d be a given finite ultrametric space with minimum non zero distance then the p negative type gap x p of x d is positive for all p 0 in this paper we compute the value of the limit x lim p x p p it turns out that this value is positive and it may be given explicitly by an elegant combinatorial formula this formula allows us to characterize when the ratio x p p is a constant independent of p the determination of x also leads to new asymptotically sharp families of enhanced p negative type inequalities for x d indeed suppose that g 0 x then for all sufficiently large p the inequality g p 2 k 1 n k 2 j i 1 n d z j z i p j i 0 holds for each finite subset z 1 z n x and each scalar n tuple 1 n r n that satisfies 1 n 0 notably these results do not extend to general finite metric spaces

on state dependent implication in quantum mechanical distant correlations let be an arbitrary given composite system state statistical operator it is shown that same subsystem events imply each other state dependently according to if and only if they act equally in the range of the corresponding subsystem state reduced statistical operator opposite subsystem events imply each other in the same way if and only if they are twin events i e if is a pure state it is shown that the anti unitary correlation operator also plays a decisive role in the latter implication

methods for constructing distance matrices and the inverse eigenvalue problem abstract let d 1 r k k and d 2 r l l be two distance matrices we provide necessary conditions on z r k l in order that d d 1 z z t d 2 r n n be a distance matrix we then show that it is always possible to border an n n distance matrix with certain scalar multiples of its perron eigenvector to construct an n 1 n 1 distance matrix we also give necessary and sufficient conditions for two principal distance matrix blocks d 1 and d 2 be used to form a distance matrix as above where z is a scalar multiple of a rank one matrix formed from their perron eigenvectors finally we solve the inverse eigenvalue problem for distance matrices in certain special cases including n 3 4 5 6 any n for which there exists a hadamard matrix and some other cases

perturbations of discrete elliptic operators abstract given v a finite set a self adjoint operator k on c v is called elliptic if it is positive semi definite and its lowest eigenvalue is simple therefore there exists a unique up to sign unitary function c v satisfying k and then k is named elliptic clearly a elliptic operator is singular iff 0 examples of elliptic operators are the so called schrodinger operators on finite connected networks as well as the signless laplacian of connected bipartite networks a elliptic operator k defines an automorphism on whose inverse is called orthogonal green operator of k we aim here at studying the effect of a perturbation of k on its orthogonal green operator the perturbation here considered is performed by adding a self adjoint and positive semi definite operator to k as particular cases we consider the effect of changing the conductances on semi definite schodinger operators on finite connected networks and on the signless laplacian of connected bipartite networks the expression obtained for the perturbed network is explicitly given in terms of the orthogonal green function of the original network

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

more on the state dependent implication in quantum mechanics the mixing structure of a given quantum state statistical operator is considered that is the pure states and associated weights of which it consists its influence on the state dependent quantum logical implication determined by is also studied it is found that it has no influence at all except through the null projector it is shown that the latter when it determines the state dependent implication in a certain sense carries with it all the boolean mathematical structure of projectors in the given boolean subalgebra of quantum logic where is the domain of definition of the state dependent implication

on q integral graphs with edge degrees at most six abstract q integral graphs are graphs whose signless laplacians have only integral eigenvalues the edge degree of an edge is the number of edges adjacent to the edge in 2008 s k simic and z stanic studied connected q integral graphs with edge degrees at most five in this paper we study connected q integral graphs with edge degrees at most six

infinite families of q integral graphs abstract in this article we characterize all signless laplacian integral graphs here called q integral graphs in the following classes complete split graphs multiple complete split like graphs extended complete split like graphs multiple extended split like graphs all these graphs were defined by hansen et al p hansen h melot d stevanovic integral complete split graphs univ beograd publ elektrotehn fak ser mat 13 2002 89 95 also we characterize all q integral graphs of the form k n 1 k n 2 k n 3 where k n i i 1 2 3 is a complete graph on n i vertices these characterizations allow us to exhibit many infinite families of q integral graphs

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

the signless laplacian coefficients and incidence energy of bicyclic graphs abstract let q g x det x i q g i 1 n 1 i i x n i be the characteristic polynomial of the signless laplacian matrix of a graph g of order n this paper investigates how the signless laplacian coefficients i e coefficients of q g x change after some graph transformations these results are used to prove that the set b n of all bicyclic graphs of order n has exactly two minimal elements with respect to the partial ordering of their coefficients furthermore we present a sharp lower bound for the incidence energy of bicyclic graphs of order n and characterize all extremal graphs

properties of euclidean and non euclidean distance matrices abstract a distance matrix d of order n is symmetric with elements 1 2 d ij 2 where dii 0 d is euclidean when the 1 2 n n 1 quantities dij can be generated as the distances between a set of n points x n p in a euclidean space of dimension p the dimensionality of d is defined as the least value of p rank x of any generating x in general p 1 and p 2 are also acceptable but may include imaginary coordinates even when d is euclidean basic properties of euclidean distance matrices are established in particular when rank d it is shown that depending on whether etd e is not or is zero the generating points lie in either p 1 dimensions in which case they lie on a hypersphere or in p 2 dimensions in which case they do not the notation e is used for a vector all of whose values are one when d is non euclidean its dimensionality p r s will comprise r real and s imaginary columns of x and r s are invariant for all generating x of minimal rank higher ranking representations can arise only from p 1 r 1 s or p 1 r s 1 or p 2 r 1 s 1 so that not only are r s invariant but they are both minimal for all admissible representations x

characterizing threshold graphs with k main signless laplacian eigenvalues abstract we show that the number k of main signless laplacian eigenvalues of a threshold graph on n 3 vertices is bounded by n 2 and we characterize for any integer 1 k n 2 all such graphs with exactly k main signless laplacian eigenvalues moreover we also determine the number of threshold graphs on n vertices with k main signless laplacian eigenvalues

spectral characterizations of signed lollipop graphs abstract let g be a signed graph where g is the underlying simple graph and e g is the sign function on the edges of g in this paper we consider the spectral characterization problem extended to the adjacency matrix and laplacian matrix of signed graphs after giving some basic results we study the spectral determination of signed lollipop graphs and we show that any signed lollipop graph is determined by the spectrum of its laplacian matrix

connected graphs with maximal q index the one dominating vertex case abstract by the signless laplacian of a simple graph g we mean the matrix q g d g a g where a g d g denote respectively the adjacency matrix and the diagonal matrix of vertex degrees of g for every pair of positive integers n k it is proved that if 3 k n 3 then h n k the graph obtained from the star k 1 n 1 by joining a vertex of degree 1 to k 1 other vertices of degree 1 is the unique connected graph that maximizes the largest signless laplacian eigenvalue over all connected graphs with n vertices and n k edges

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

approximation by matrices positive semidefinite on a subspace abstract we obtain the best approximation to a matrix by matrices positive semidefinite on a subspace as a by product we present two new characterizations of euclidean distance matrices

permanental roots and the star degree of a graph abstract for undirected graphs without loops or multiple edges we define the star degree of a graph and prove that it is equal to the multiplicity of the root 1 of per xi b where b d a considering bipartite graphs we prove that per xi b per xi l where l d a and consequently that the star degree of a bipartite graph can also be characterized by the multiplicity of the root 1 of per xi l

signless laplacian spectral radius and hamiltonicity we give tight conditions on the signless laplacian spectral radius of a graph for the existence of hamiltonian paths and cycles

supremal p negative type of vertex transitive graphs abstract we study the supremal p negative type of connected vertex transitive graphs the analysis provides a way to characterize subsets of the hamming cube 0 1 n l 1 n n 1 that have strict 1 negative type the result can be stated in two ways a subset s x 0 x 1 x k of the hamming cube 0 1 n l 1 n has generalized roundness one if and only if the vectors x 1 x 0 x 2 x 0 x k x 0 are linearly dependent in r n equivalently s has strict 1 negative type if and only if the vectors x 1 x 0 x 2 x 0 x k x 0 are linearly independent in r n

on the supremal p negative type of finite metric spaces abstract we study the supremal p negative type of finite metric spaces an explicit expression for the supremal p negative type x d of a finite metric space x d is given in terms of its associated distance matrix from which the supremal p negative type of the space may be calculated the method is then used to give a straightforward calculation of the supremal p negative type of the complete bipartite graphs k n m endowed with the usual path metric a gap in the spectrum of possible supremal p negative type values of path metric graphs is also proven

finite metric spaces of strictly negative type abstract we prove that if a finite metric space is of strictly negative type then its transfinite diameter is uniquely realized by the infinite extender load vector finite metric spaces that have this property include all spaces on two three or four points all trees and all finite subspaces of euclidean spaces we prove that if the distance matrix is both hypermetric and regular then it is of strictly negative type we show that the strictly negative type finite subspaces of spheres are precisely those which do not contain two pairs of antipodal points in connection with an open problem raised by kelly we conjecture that all finite subspaces of hyperbolic spaces are hypermetric and regular and hence of strictly negative type

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

the cone of distance matrices abstract the geometry of the cone of euclidean distance matrices edms is analyzed using a new characterization of an edm the facial structure and the angle that edms of embedding dimension one make with the center ray are found this result follows from a complete analysis of the critical points of the distance function in frobenius norm from the matrix e consisting of zero diagonal and ones elsewhere to the edms of embedding dimension one

on the signless laplacian spectral radius of graphs with cut vertices in this paper we show that among all the connected graphs with n vertices and k cut vertices the maximal signless laplacian spectral radius is attained uniquely at the graph gn k where gn k is obtained from the complete graph kn k by attaching paths of almost equal lengths to all vertices of kn k we also give a new proof of the analogous result for the spectral radius of the connected graphs with n vertices and k cut vertices see a berman x d zhang on the spectral radius of graphs with cut vertices j combin theory ser b 83 2001 233 240 finally we discuss the limit point of the maximal signless laplacian spectral radius