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

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

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

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

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

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

free vibration analysis of laminated orthotropic spherical shells abstract analytical solutions are presented for the free vibration of laminated orthotropic spherical shells the analysis is based on the application of the chebyshev galerkin spectral method for the evaluation of free vibration frequencies and mode shapes the eigenvalue problem is defined in terms of five equations of motion in terms of the meridional coordinate the circumference variation of the displacements being represented in a fourier series numerical results are presented to show the validity and accuracy of the proposed method results of parametric studies for laminated annular spherical shells with different orthotropic ratios opening angles lamination sequences and shear deformation are also presented

autour de la conjecture de l markus sur les varietes affines for any subgroupg of n we introduce some integer discg n called thediscompacity ofg this number measures to what extent the closure ofg is not compact the markus conjecture says that a compact affinely flat unimodular manifold is complete our main result called the discompact theorem is that this conjecture is true under the assumption that the linear holonomy i e the parallel transport has discompacity 1 because discso n 1 1 1 this ensures that a compact flat lorentz manifoldm is geodesically complete hence by a previous result of w goldman and y kamishima gk such am is up to finite covering a solvmanifold this achieves the proof of a bieberbach s theorem for compact lorentz flat manifolds

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

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

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

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

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

nonlinear analysis of anisotropic panels a computational procedure is presented for the efficient nonlinear analysis of symmetric anisotropic panels the three key elements of the procedure are 1 use of three field mixed models having independent interpolation shape functions for stress resultants strain components and generalized displacements with the stress resultants and strain components allowed to be discontinuous at interelement boundaries 2 decomposition of the material stiffness matrix into the sum of an orthotropic and a nonorthotropic anisotropic part and 3 successive application of the finite element method and the classical rayleigh ritz technique the finite element method is first used to generate a few global approximation vectors or modes then the amplitudes of these modes are computed by using the rayleigh ritz technique the global approximation vectors are taken to be various order derivatives of the strain components stress resultants and generalized displacements with respect to an anisotropic tracing parameter and a load parameter they are evaluated at zero values of the two parameters the size of the analysis model used in generating the global approximation vectors is identical to that of the corresponding orthotropic structure the effectiveness of the proposed computational procedure is demonstrated by means of a numerical example and its potential for solving quasi symmetric nonlinear problems of composite structures is discussed

some global properties of static spacetimes we show that a static lorentzian manifold satisfying a completeness assumption is geodesically connected in particular such condition is satisfied by all compact static manifolds in the compact case we also prove the existence of a closed geodesic in every free homotopy class determined by a finite number of deck transformations

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

mixed models and reduced selective integration displacement models for nonlinear shell analysis simple mixed models are developed for the geometrically nonlinear analysis of shells a total lagrangian description of the shell deformation is used and the analytical formulation is based on a form of the nonlinear shallow shell theory with the effects of transverse shear deformation and bending extensional coupling included the fundamental unknowns consist of eight stress resultants and five generalized displacements of the shell and the element characteristic arrays are obtained by using the hellinger reissner mixed variational principle the polynomial interpolation or shape functions used in approximating the stress resultants are in general of different degree than those used for approximating the generalized displacements the stress resultants are discontinuous at the element boundaries and are eliminated on the element level the equivalence and near equivalence between the mixed models developed herein and displacement models based on reduced selective integration of both transverse shear and extensional energy terms is discussed the use of reduction methods in conjunction with the mixed models is outlined and the advantages of mixed models over displacement models are delineated analytic expressions are derived for the rigid body and spurious or zero energy models for the various mixed models and their equivalent displacement models also the advantages of mixed models over equivalent displacement models are outlined numerical results are presented to demonstrate the high accuracy and effectiveness of the mixed models developed and to compare their performance with other mixed models reported in the literature

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

lorentzian covering space and homotopy classes we analyze structures of covering space over a lorentzian manifold by use of this we show that if a lorentzian manifold is globally hyperbolic then for any two causally related points p and q the number of homotopy classes of causal curves from p to q is finite and each of its homotopy classes has a causal geodesic from p to q

structure of lorentzian tori with a killing vector field all lorentzian tori with a non discrete group of isometries are characterized and explicitly obtained they can lie into three cases a flat b conformally flat but non flat and c geodesically incomplete a detailed study of many of their properties including results on the logical dependence of the three kinds of causal completeness on geodesic connectedness and on prescribed curvature is carried out the incomplete case is specially analyzed and several known examples and results in the literature are generalized from a unified point of view

closed timelike geodesics on montre que chaque classe de t homotopie stable libre de courbes fermees du genre temps dans une variete de lorentz compacte contient une courbe la plus longue qui doit etre une geodesique fermee du genre temps

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

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

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

closed timelike geodesics in compact spacetimes let m be a compact spacetime which admits a regular globally hyperbolic covering and c a nontrivial free timelike homotopy class of closed timelike curves in m we prove that c contains a longest curve which must be a closed timelike geodesic if and only if the timelike injectivity radius of c is finite i e c has a bounded length as a consequence among others we deduce that for a compact static spacetime there exists a closed timelike geodesic within every nontrivial free timelike homotopy class having a finite timelike injectivity radius

closed geodesies on compact lorentzian manifolds of splitting type in this paper we consider the problem of the existence of a spacelike closed geodesic on compact lorentzian manifolds tipler and galloway proved that under suitable topological properties of the manifold there exists a closed timelike geodesic in their proofs they use the hypothesis that the time coordinate of one timelike geodesic has derivative always different from zero this clearly fails for spacelike geodesies using variational methods and applying the relative category theory we prove the existence of a closed spacelike geodesic on a compact manifold of splitting type observe that thanks to the previous results the existence of at least two geometrically distinct closed geodesies on follows

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

nonlinear theory for plates and shells including the effects of transverse shearing nonlinear strain displacement relations for three dimensional elasticity are determined in orthogonal curvilinear coordinates to develop a two dimensiona l theory the displacements are expressed by trigonometric series representation through the thickness the nonlinear strain displacement relations are expanded into a series that contains all first and second degree terms in the series for the displacements only the first few terms are retained insertion of the expansions into the three dimensional virtual work expression leads to nonlinear equations of equilibrium for laminated and thick plates and shells that include the effects of transverse shearing equations of equilibrium and buckling equations are derived for flat plates and cylindrical shells the shell equations reduce to conventional transverse shearing shell equations when the effects of the trigonometric terms are omitted and to classical shell equations when the trigonometric terms are omitted and the shell is assumed to be thin numerical results are presented for the buckling of a thick simply supported flat rectangular plate in longitudinal compression

a note on the chebyshev coefficients of the general order derivative of an infinitely differentiable function abstract a formula expressing the chebyshev coefficients of the general order derivative of an infinitely differentiable function in terms of its chebyshev coefficients is suggested

nonlinear analysis of laminated axisymmetric spherical shells abstract the governing differential equations for the problem of laminated axisymmetric spherical shells undergoing large deformations are formulated using the principle of virtual work an analytical solution of the governing equations based on the chebyshev galerkin spectral method is investigated the efficacy and applicability of the solution procedure is discovered using numerical results parametric studies are conducted to bring out the effect of factors like orthotropy ratio r h ratio shear deformation and opening angle on the large deflection behaviour of laminated orthotropic spherical shells and interesting observations are made the numerical results should prove helpful in testing the nonlinear composite shell finite elements

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

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

closed geodesics in lorentzian surfaces we introduce class a spacetimes i e compact vicious spacetimes m g such that the abelian cover m g is globally hyperbolic we study the main properties of class a spacetimes using methods similar to the one introduced in 19 and 3 as a consequence we are able to characterize manifolds admitting class a metrics completely as mapping tori the set of class a spacetimes is shown to be open in the c0 topology on the set of lorentzian metrics as an application we prove a coarse lipschitz property for the time separation of the abelian cover

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

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

connecting solutions of the lorentz force equation do exist recent results on the maximization of the charged particle action open image in new window in a globally hyperbolic spacetime are discussed and generalized we focus on the maximization of open image in new window over a given causal homotopy class open image in new window of curves connecting two causally related events x0 x1 action open image in new window is proved to admit a maximum on open image in new window and also one in the adherence of each timelike homotopy class c moreover the maximum 0 on open image in new window is timelike if open image in new window contains a timelike curve and the degree of differentiability of all the elements is at least c2

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

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

on the existence of closed timelike geodesics in compact spacetimes in 19 tipler has shown that a compact spacetime having a regular globally hyperbolic covering space with compact cauchy surfaces necessarily contains a closed timelike geodesic the restriction to compact spacetimes with just regular globally hyperbolic coverings i e the cauchy surfaces are not required to be compact is still an open question here we shall answer this question negatively by providing examples of compact flat lorentz space forms without closed timelike geodesics and shall give some criterion for the existence of such geodesics more generally we will show that in a compact spacetime having a regular globally hyperbolic covering each free timelike homotopy class determined by a central deck transformation must contain a closed timelike geodesic whether or not a compact flat spacetime contains closed nonspacelike geodesics is as far as we know an open question we shall answer this question affirmatively we shall also introduce the notion of timelike injectivity radius for a spacetime relative to a free timelike homotopy class and shall show that it is finite whenever the corresponding deck transformation is central

periodic geodesics and geometry of compact lorentzian manifolds with a killing vector field we study the geometry and the periodic geodesics of a compact lorentzian manifold that has a killing vector field which is timelike somewhere using a compactness argument for subgroups of the isometry group we prove the existence of one timelike non self intersecting periodic geodesic if the killing vector field is nowhere vanishing then there are at least two distinct periodic geodesics as a special case compact stationary manifolds have at least two periodic timelike geodesics we also discuss some properties of the topology of such manifolds in particular we show that a compact manifold m admits a lorentzian metric with a nowhere vanishing killing vector field which is timelike somewhere if and only if m admits a smooth circle action without fixed points

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

isometries of lorentz surfaces and convergence groups we study the isometry group of a globally hyperbolic spatially compact lorentz surface such a group acts on the circle and we show that when the isometry group acts non properly the subgroups of mathrm diff mathbb s 1 obtained are semi conjugate to subgroups of finite covers of mathrm psl 2 mathbb r by using convergence groups under an assumption on the conformal boundary we show that we have a conjugacy in mathrm homeo mathbb s 1

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