products of involutions in o v abstract let v be a finite dimensional vector space over a commutative field of characteristic distinct from 2 let v carry a symmetric nondegenerate bilinear form the special orthogonal group is o v o v det 1 main result an element o v is a product of two involutions in o v ifand only if dim v 2 mod 4 or an orthogonal decomposition of v into orthogonally indecomposable modules contains a module of odd dimension

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

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

an algebraic characterization of vacuum states in minkowski space iii reflection maps employing the algebraic framework of local quantum physics vacuum states in minkowski space are distinguished by a property of geometric modular action this property allows one to construct from any locally generated net of observables and corresponding state a continuous unitary representation of the proper poincare group which acts covariantly on the net and leaves the state invariant the present results and methods substantially improve upon previous work in particular the continuity properties of the representation are shown to be a consequence of the net structure and surmised cohomological problems in the construction of the representation are resolved by demonstrating that for the poincare group continuous reflection maps are restrictions of continuous homomorphisms

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

normal forms of symplectic pencils and the discrete time algebraic riccati equation abstract the solution of the discrete time algebraic riccati equation leads to symplectic pencils of matrices normal forms of such pencils under symplectic equivalence are determined special attention is given to characteristic roots of modulus 1 and their corresponding elementary divisors and inertial invariants

anti triangular and anti m hessenberg forms for hermitian matrices and pencils hermitian pencils i e pairs of hermitian matrices arise in many applications such as linear quadratic optimal control or quadratic eigenvalue problems we derive conditions from which anti triangular and anti m hessenberg forms for general including singular hermitian pencils can be obtained under unitary equivalence transformations

conjugacy classes of involutions in the lorentz group ω v and in so v abstract the lorentz group v is bireflectional and all involutions in v are conjugate more generally we give conditions for two involutions to be conjugate in so v provided that v is a vector space over a finite field or over an ordered field

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

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

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

a schur decomposition for hamiltonian matrices abstract a schur type decomposition for hamiltonian matrices is given that relies on unitary symplectic similarity transformations these transformations preserve the hamiltonian structure and are numerically stable making them ideal for analysis and computation using this decomposition and a special singular value decomposition for unitary symplectic matrices a canonical reduction of the algebraic riccati equation is obtained which sheds light on the sensitivity of the nonnegative definite solution after presenting some real decompositions for real hamiltonian matrices we look into the possibility of an orthogonal symplectic version of the qr algorithm suitable for hamiltonian matrices a finite step initial reduction to a hessenberg type canonical form is presented however no extension of the francis implicit shift technique was found and reasons for the difficulty are given

normal forms of elements of classical real and complex lie and jordan algebras elements of the classical complex and real lie and jordan algebras with involutions are classified into conjugacy classes under the action of the corresponding classical lie group normal forms of representatives of each conjugacy class are chosen so as to resemble the jordan normal forms of n n complex matrices for completeness similar results are given for gl n c gl n r and gl n h

pencils of complex and real symmetric and skew matrices abstract this expository paper establishes the canonical forms under congruence for pairs of complex or real symmetric or skew matrices the treatment is in the spirit of the well known book of gantmacher on matrix theory and may be regarded as a supplement to gantmacher s chapters on pencils of matrices

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

existence uniqueness and stability of solutions to singular linear quadratic optimal control problems we consider general continuous and discrete time linear quadratic optimal control problems where the dynamics of the system is a linear differential or difference algebraic equation with constant coefficient matrices and the cost functionals are defined by infinite integrals or sums these problems occur in numerous applications in many different fields such as economics operations research electrical engineering model and system design biology and chemistry since they form two of the simplest optimal control problems and they are also used as linear approximations for many nonlinear problems which are much harder to solve we extend some of the known results to these general problems

on hamiltonian and symplectic hessenberg forms abstract characterizations are given for the hamiltonian matrices that can be reduced to hamiltonian hessenberg form and the symplectic matrices that can be reduced to symplectic hessenberg form by orthogonal symplectic similarity transformations the reduction to these special hessenberg forms is the missing link in the solution of the open problem of constructing a stable structure preserving qr like method of complexity o n 3 for the computation of invariant subspaces of hamiltonian and symplectic matrices our considerations lead us to propose an approach to the computation of lagrangian invariant subspaces of a hamiltonian or symplectic matrix

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

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

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

canonical forms for hamiltonian and symplectic matrices and pencils abstract we study canonical forms for hamiltonian and symplectic matrices or pencils under equivalence transformations which keep the class invariant in contrast to other canonical forms our forms are as close as possible to a triangular structure in the same class we give necessary and sufficient conditions for the existence of hamiltonian and symplectic triangular jordan kronecker and schur forms the presented results generalize results of lin and ho w w lin t c ho on schur type decompositions for hamiltonian and symplectic pencils technical report institute of applied mathematics national tsing hua university taiwan 1990 and simplify the proofs presented there

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

numerical solutions for large sparse quadratic eigenvalue problems abstract we study the quadratic eigenvalue problem a b 2c x 0 where a b and c are symmetric real n n matrices and a c are positive definite we propose an efficient numerical algorithm to compute a few of the smallest positive eigenvalues of the problem and their associated eigenvectors the new algorithm includes two parts the first part gives iterative methods which can be used to compute the smallest positive eigenvalue we develop a globally linearly convergent basic iteration and two locally quadratically convergent iterations the second part uses the nonequivalence deflation technique this technique allows us to transform the original problem to a new problem with different a b and c the new problem has the same eigenvalues as the old problem except that the smallest positive eigenvalue of the old problem is replaced by zero therefore the second smallest positive eigenvalue of the old problem becomes the smallest positive one for the new problem then the above proposed iterative methods can be applied again to find the second smallest positive eigenvalue proceeding in this way we can find out the 3rd 4th smallest positive eigenvalues our algorithm utilizes the symmetry and positivity of the given matrices and avoids computing the undesired complex conjugate eigenvalues under some mild conditions is efficient and reliable the above process can also be used to find a few of the largest negative eigenvalues

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

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

a conjecture on the diameter and signless laplacian index of graphs abstract a bug bug p q 1 q 2 is a graph obtained from a complete graph k p by deleting an edge uv and attaching paths p q 1 and p q 2 at u and v respectively in this paper we show that for connected graphs g of order n with signless laplacian index q 1 g and diameter diam g q 1 g diam g is maximized for and only for the graph bug n 2 2 p q where p d 2 q d 2 and d n 1 2 this solves a conjecture in 6 on the signless laplacian index involving the diameter

on doubly structured matrices and pencils that arise in linear response theory abstract we discuss matrix pencils with a double symmetry structure that arise in the hartree fock model in quantum chemistry we derive anti triangular condensed forms from which the eigenvalues can be read off ideally these would be condensed forms under unitary equivalence transformations that can be turned into stable structure preserving numerical methods for the pencils under consideration this is a difficult task and not always possible we present necessary and sufficient conditions when this is possible if this is not possible then we show how we can include other transformations that allow to reduce the pencil to an almost anti triangular form