novel inequalities for generalized graph entropies graph energies and topological indices the entropy of a graph is an information theoretic quantity for measuring the complexity of a graph after shannon introduced the entropy to information and communication many generalizations of the entropy measure have been proposed such as renyi entropy and daroczy entropy in this article we prove accurate connections inequalities between generalized graph entropies graph energies and topological indices additionally we obtain some extremal properties of nine generalized graph entropies by employing graph energies and topological indices

bicyclic graphs with small positive index of inertia abstract the positive index of inertia of a graph g denoted by i g is the number of the positive eigenvalues of the adjacency matrix of g in this paper we investigate the minimal positive index of inertia among all bicyclic graphs of order n with pendant vertices and characterize the bicyclic graphs with positive index 1 or 2 among all bicyclic graphs of order n

a hamiltonian preserving discontinuous galerkin method for the generalized korteweg de vries equation the invariant preserving property is one of the guiding principles for numerical algorithms in solving wave equations in order to minimize phase and amplitude errors after long time simulation in this paper we design analyze and numerically validate a hamiltonian preserving discontinuous galerkin method for solving the korteweg de vries kdv equation for the generalized kdv equation the semi discrete formulation is shown to preserve both the first and the third conserved integrals and approximately preserve the second conserved integral for the linearized kdv equation all the first three conserved integrals are preserved and optimal error estimates are obtained for polynomials of even degree the preservation properties are also maintained by the fully discrete dg scheme our numerical experiments demonstrate both high accuracy of convergence and preservation of all three conserved integrals for the generalized kdv equation we also show that the shape of the solution after long time simulation is well preserved due to the hamiltonian preserving property

self similar solutions to a density dependent reaction diffusion model in this paper we investigated a density dependent reaction diffusion equation u t u m xx u u m this equation is known as the extension of the fisher or kolmogoroff petrovsky piscounoff equation which is widely used in population dynamics combustion theory and plasma physics by employing a suitable transformation this equation was mapped to the anomalous diffusion equation where the nonlinear reaction term was eliminated due to its simpler form some exact self similar solutions with compact support have been obtained the solutions evolving from an initial state converge to the usual traveling wave at a certain transition time hence the connection between the self similar solution and the traveling wave solution is quite clear from these results moreover the solutions were found in a manner that propagates either to the right or to the left furthermore the two solutions form a symmetric solution expanding in both directions applications to spatiotemporal pattern formation in biological populations is discussed

information processing in complex networks graph entropy and information functionals this paper introduces a general framework for defining the entropy of a graph our definition is based on a local information graph and on information functionals derived from the topological structure of a given graph more precisely an information functional quantifies structural information of a graph based on a derived probability distribution such a probability distribution leads directly to an entropy of a graph then the structural information content of a graph will be is interpreted and defined as the derived graph entropy another major contribution of this paper is the investigation of relationships between graph entropies in addition to this we provide numerical results demonstrating not only the feasibility of our method which has polynomial time complexity but also its usefulness with regard to practical applications aiming to an understanding of information processing in complex networks

unicyclic graphs with large energy we study the energy i e the sum of the absolute values of all eigenvalues of so called tadpole graphs which are obtained by joining a vertex of a cycle to one of the ends of a path by means of the coulson integral formula and careful estimation of the resulting integrals we prove two conjectures on the largest and second largest energy of a unicyclic graph due to caporossi cvetkovic gutman and hansen and gutman furtula and hua respectively moreover we characterise the non bipartite unicyclic graphs whose energy is largest

upper bounds for some graph energies a general inequality for non negative real numbers is proven based on it upper bounds for ordinary graph energy minimum dominating energy minimum covering energy laplacian energy like invariant laplacian energy randic energy and incidence energy are obtained

regular article a fast spectral algorithm for nonlinear wave equations with linear dispersion spectral algorithms offer very high spatial resolution for a wide range of nonlinear wave equations on periodic domains including well known cases such as the korteweg de vries and nonlinear schrodinger equations for the best computational efficiency one needs also to use high order methods in time while somehow bypassing the usual severe stability restrictions we use linearly implicit multistep methods with the innovation of choosing different methods for different ranges in fourier space high accuracy at low wavenumbers and a stability at high wavenumbers this new approach compares favorably to alternatives such as split step and integrating factor or linearly exact methods

the pseudospectral method for third order differential equations generalized quadrature rules are derived which assist in the selection of collocation points for the pseudospectral solution of differential equations in particular it is shown that for an nth order differential equation in one space dimension with two point derivative boundary conditions an ideal choice of interior collocation points is the set of zeros of a jacobi polynomial the pseudospectral solution of a third order initial boundary value problem is considered and accuracy is assessed by examining how well the discrete eigenproblem approximates the continuous one convergence is established for a special choice of collocation points and numerical results are included to demonstrate the viability of the approach

hermite spectral methods with a time dependent scaling for parabolic equations in unbounded domains hermite spectral methods are investigated for linear diffusion equations and nonlinear convection diffusion equations in unbounded domains when the solution domain is unbounded the diffusion operator no longer has a compact resolvent which makes the hermite spectral methods unstable to overcome this difficulty a time dependent scaling factor is employed in the hermite expansions which yields a positive bilinear form as a consequence stability and spectral convergence can be established for this approach the present method plays a similar role in the stability of the similarity transformation technique proposed by funaro and kavian math comp 57 1991 pp 597 619 however since coordinate transformations are not required the present approach is more efficient and is easier to implement in fact with the time dependent scaling the resulting discretization system is of the same form as that associated with the classical straightforward but unstable hermite spectral method numerical experiments are carried out to support the theoretical stability and convergence results

staggered time integrators for wave equations we consider variations of the adams bashforth backward differentiation and runge kutta families of time integrators to solve systems of linear wave equations on uniform time staggered grids these methods are found to have smaller local truncation errors and to allow larger stable time steps than traditional nonstaggered versions of equivalent orders we investigate the accuracy and stability of these methods analytically experimentally and through the use of a novel root portrait technique

the locally extrapolated exponential time differencing lod scheme for multidimensional reaction diffusion systems this paper introduces the local extrapolation of first order locally one dimensional exponential time differencing scheme for numerical solution of multidimensional nonlinear reaction diffusion systems this novel scheme has the benefit of solving multidimensional problems in locally one dimensional fashion by implementing sequences of tridiagonal matrix solvers instead of solving a banded system the storage size needed for solving systems in higher dimensions with this scheme is similar to that needed for one spatial dimension systems the stability monotonicity and convergence of the locally extrapolated exponential time differencing scheme have been examined stability analysis shows that the scheme is strongly stable l stable and is particularly beneficial to nonlinear partial differential equations with irregular initial data or discontinuity involving initial and boundary conditions due to its ability to damp spurious oscillations caused by high frequency components in the solution the performance of the novel scheme has been investigated by testing it on a two dimensional schnakenberg model two and three dimensional brusselator models and a three dimensional enzyme kinetics of michaelis menten type reaction diffusion model numerical experiments demonstrate the efficiency accuracy and reliability of the scheme locally extrapolated lod exponential time differencing scheme is developed multidimensional nonlinear reaction diffusion systems are considered monotonicity and stability are analyzed the order of convergence is examined

some practical experience with the time integration of dissipative equations different methods for the numerical integration of evolution dissipative partial differential equations are tested with the kuramoto sivashinsky equation discretizations in space include galerkin and nonlinear galerkin methods for integration in time three different codes are used including standard stiff ode methods numerical tests show that standard codes for stiff ode render a gain of computing time of several orders of magnitude with respect problem tailored methods

some unconditionally stable time stepping methods for the 3d maxwell s equations almost all the difficulties that arise in the numerical solution of maxwell s equations are due to material interfaces in case that their geometrical features are much smaller than a typical wavelength one would like to use small space steps with large time steps the first time stepping method which combines a very low cost per time step with unconditional stability was the adi fdtd method introduced in 1999 the present discussion starts with this method and with an even more recent crank nicolson based split step method with similar properties we then explore how these methods can be made even more efficient by combining them with techniques that increase their temporal accuracies

on graph entropy measures based on the number of independent sets and matchings abstract in this paper we consider the graph entropy measures based on the number of independent sets and matchings the reason to study these measures relates to the fact that the independent set and matching problem is computationally demanding however these features can be calculated for smaller networks in case one can establish efficient estimations those measures may be also used for larger graphs so we establish some upper and lower bounds as well as some information inequalities for these information theoretic quantities in order to give further evidence we also generate numerical results to study these measures such as list the extremal graphs for these entropies those results reveal the two entropies possess some new features

fourth order time stepping for stiff pdes a modification of the exponential time differencing fourth order runge kutta method for solving stiff nonlinear pdes is presented that solves the problem of numerical instability in the scheme as proposed by cox and matthews and generalizes the method to nondiagonal operators a comparison is made of the performance of this modified exponential time differencing etd scheme against the competing methods of implicit explicit differencing integrating factors time splitting and fornberg and driscoll s sliders for the kdv kuramoto sivashinsky burgers and allen cahn equations in one space dimension implementation of the method is illustrated by short matlab programs for two of the equations it is found that for these applications with fixed time steps the modified etd scheme is the best

the galerkin method for the kdv equation using a new basis of smooth piecewise cubic polynomials abstract in this paper we introduce the galerkin method using a new basis of smooth piecewise cubic polynomials for the one dimensional nonlinear kdv equation the stability analysis shows that the present method is unconditionally stable in l 2 norm numerical experiments show that the proposed method preserves the conservation laws and is effective for simulating the motions and the interactions of solitary waves

graph operations based on using distance based graph entropies abstract let g be a connected graph the eccentricity of vertex v is the maximum distance between v and other vertices of g in this paper we study a new version of graph entropy based on eccentricity of vertices of g in continuing we study this graph entropy for some classes of graph operations finally we compute the graph entropy of two classes of fullerene graphs

the skew energy of a digraph we are interested in the energy of the skew adjacency matrix of a directed graph d which is simply called the skew energy of d in this paper properties of the skew energy of d are studied in particular a sharp upper bound for the skew energy of d is derived in terms of the order of d and the maximum degree of its underlying undirected graph an infinite family of digraphs attaining the maximum skew energy is constructed moreover the skew energy of a directed tree is independent of its orientation and interestingly it is equal to the energy of the underlying undirected tree skew energies of directed cycles under different orientations are also computed some open problems are presented

efficient spectral galerkin method i direct solvers of second and fourth order equations using legendre polynomials this paper presents some efficient algorithms based on the legendre galerkin approximations for the direct solution of the second and fourth order elliptic equations the key to the efficiency of these algorithms is to construct appropriate base functions which lead to systems with sparse matrices for the discrete variational formulations the complexities of the algorithms are a small multiple of n d 1 operations for a d dimensional domain with n 1 d unknowns while the convergence rates of the algorithms are exponential for problems with smooth solutions in addition the algorithms can be effectively parallelized since the bottlenecks of the algorithms are matrix matrix multiplications

on the zeros of the partial hosoya polynomial of graphs abstract the partial hosoya polynomial or briefly the partial h polynomial can be used to construct the well known hosoya polynomial the ith coefficient of this polynomial defined for an arbitrary vertex u of a graph g is the number of vertices at distance i from u the aim of this paper is to determine the partial h polynomial of several well known graphs and then to investigate the location of their zeros to pursue we characterize the structure of graphs with the minimum and the maximum modulus of the zeros of partial h polynomial finally we define another graph polynomial of the partial h polynomial see 9 also we determine the unique positive root of this polynomial for particular graphs

the fourier pseudospectral method with a restrain operator for the korteweg de vries equation abstract in this paper we develop a fourier pseudospectral method with a restrain operator for the korteweg de vries equation we prove the generalized stability of the schemes and give convergence estimations depending on the smoothness of the solution of the p d e

a legendre petrov galerkin method for fourth order differential equations in this paper we present a legendre petrov galekin method for one dimensional linear fourth order differential equations a legendre petrov galerkin and chebyshev collocation method is developed for the nonlinear kuramoto sivashinsky equation numerical results are presented to demonstrate the efficiency of the proposed schemes and optimal rates of convergence in the l 2 norm are rigorously derived

a method to compute statistics of large noise induced perturbations of nonlinear schrodinger solitons we describe in detail the application of importance sampling to numerical simulations of large noise induced perturbations in soliton based optical transmission systems governed by the nonlinear schrodinger equation the method allows one to concentrate the samples in monte carlo simulations around those noise realizations that are most likely to produce the large pulse deformations connected with errors and it yields computational speedups of several orders of magnitude over standard monte carlo simulations we demonstrate the method by using it to calculate the probability density functions associated with pulse amplitude frequency timing and phase fluctuations in a prototypical soliton based communication system

optimal error estimates of the legendre petrov galerkin method for the korteweg de vries equation in this paper the legendre petrov galerkin method for the korteweg de vries equation with nonperiodic boundary conditions is analyzed the nonlinear term is computed with the legendre spectral method and some pseudospectral methods respectively optimal error estimates in l2 norm are obtained for both semidiscrete and fully discrete schemes the method is also applicable to some 2m 1 th order differential equations

convergence analysis of spectral collocation methods for a singular differential equation solutions of partial differential equations with coordinate singularities often have special behavior near the singularities which forces them to be smooth special treatment for these coordinate singularities is necessary in spectral approximations in order to avoid degradation of accuracy and efficiency it has been observed numerically in the past that for a scheme to attain high accuracy it is unnecessary to impose all the pole conditions the constraints representing the special solution behavior near singularities in this paper we provide a theoretical justification for this observation specifically we consider an existing approach which uses a pole condition as the boundary condition at a singularity and solves the reformulated boundary value problem with a commonly used gauss lobatto collocation scheme spectral convergence of the legendre and chebyshev collocation methods is obtained for a singular differential equation arising from polar and cylindrical geometries

analytical and numerical aspects of certain nonlinear evolution equations ii numerical nonlinear schrodinger equation abstract various numerical methods are employed in order to approximate the nonlinear schrodinger equation namely i the classical explicit method ii hopscotch method iii implicit explicit method iv crank nicolson implicit scheme v the ablowitz ladik scheme vi the split step fourier method f tappert and vii pseudospectral fourier method fornberg and whitham comparisons between the ablowitz ladik scheme which was developed using notions of the inverse scattering transform and the other utilized schemes are obtained

optimal error estimates in jacobi weighted sobolev spaces for polynomial approximations on the triangle spectral approximations on the triangle by orthogonal polynomials are studied in this paper optimal error estimates in weighted semi norms for both the l2 and h 1 0 orthogonal polynomial projections are established by using the generalized koornwinder polynomials and the properties of the sturm liouville operator on the triangle these results are then applied to derive error estimates for the spectral galerkin method for second and fourth order equations on the triangle the generalized koornwinder polynomials and approximation results developed in this paper will be useful for many other applications involving spectral and spectral element approximations in triangular domains

relations between distance based and degree based topological indices let w sz pi and wp be respectively the wiener szeged pi and wiener polarity indices of a molecular graph g let m1 and m2 be the first and second zagreb indices of g we obtain relations between these classical distance and degree based topological indices

error analysis and applications of the fourier galerkin runge kutta schemes for high order stiff pdes an integrating factor mixed with runge kutta technique is a time integration method that can be efficiently combined with spatial spectral approximations to provide a very high resolution to the smooth solutions of some linear and nonlinear partial differential equations in this paper the novel hybrid fourier galerkin runge kutta scheme with the aid of an integrating factor is proposed to solve nonlinear high order stiff pdes error analysis and properties of the scheme are provided application to the approximate solution of the nonlinear stiff korteweg de vries the 3rd order pde dispersive equation kuramoto sivashinsky the 4th order pde dissipative equation and kawahara the 5th order pde equations are presented comparisons are made between this proposed scheme and the competing method given by kassam and trefethen it is found that for kdv ks and kawahara equations the proposed method is the best

fisher equation with density dependent diffusion special solutions a special family of solutions explicit in the space and time variables is found for the fisher equation with density dependent diffusion the connection with the known travelling wave solution and the initial conditions from which that evolves is also shown

a hybrid numerical method for the kdv equation by finite difference and sinc collocation method abstract in this paper we propose a hybrid numerical method for the kdv equation more precisely we discretize the temporal derivative of kdv equation by a weighted scheme and treat the implicitly nonlinear term with the combination of finite difference and sinc collocation method the stability analysis is presented and numerical experiments illustrate the efficiency and stability of the proposed hybrid method the motions and the interactions of solitary waves relying on particular initial boundary conditions are also simulated

complete solution to a problem on the maximal energy of unicyclic bipartite graphs abstract the energy of a simple graph g denoted by e g is defined as the sum of the absolute values of all eigenvalues of its adjacency matrix denote by c n the cycle and p n 6 the unicyclic graph obtained by connecting a vertex of c 6 with a leaf of p n 6 caporossi et al conjectured that the unicyclic graph with maximal energy is p n 6 for n 8 12 14 and n 16 in hou et al 2002 y hou i gutman c woo unicyclic graphs with maximal energy linear algebra appl 356 2002 27 36 the authors proved that e p n 6 is maximal within the class of the unicyclic bipartite n vertex graphs differing from c n and they also claimed that the energies of c n and p n 6 is quasi order incomparable and left this as an open problem in this paper by utilizing the coulson integral formula and some knowledge of real analysis especially by employing certain combinatorial techniques we show that the energy of p n 6 is greater than that of c n for n 8 12 14 and n 16 which completely solves this open problem and partially solves the above conjecture

a composite runge kutta method for the spectral solution of semilinear pdes a new composite runge kutta rk method is proposed for semilinear partial differential equations such as korteweg de vries nonlinear schrodinger kadomtsev petviashvili kp kuramoto sivashinsky ks cahn hilliard and others having high order derivatives in the linear term the method uses fourier collocation and the classical fourth order rk method except for the stiff linear modes which are treated with a linearly implicit rk method the composite rk method is simple to implement indifferent to the distinction between dispersive and dissipative problems and as efficient on test problems for ks and kp as any other generally applicable method

a local discontinuous galerkin method for the korteweg de vries equation with boundary effect a local discontinuous galerkin method for solving korteweg de vries kdv type equations with non homogeneous boundary effect is developed we provide a criterion for imposing appropriate boundary conditions for general kdv type equations the discussion is then focused on the kdv equation posed on the negative half plane which arises in the modeling of transition dynamics in the plasma sheath formation h liu m slemrod kdv dynamics in the plasma sheath transition appl math lett 17 4 2004 401 410 the guiding principle for selecting inter cell fluxes and boundary fluxes is to ensure the l 2 stability and to incorporate given boundary conditions the local discontinuous galerkin method thus constructed is shown to be stable and efficient numerical examples are given to confirm the theoretical result and the capability of this method for capturing soliton wave phenomena and various boundary wave patterns

counterexamples to conjectures on graph distance measures based on topological indexes in this paper we disprove three conjectures from dehmer et al 2014 on graph distance measures based on topological indices by providing explicit classes of trees that do not satisfy proposed inequalities the constructions are based on the families of trees that have the same wiener index graph energy or randic index but different degree sequences

extremal laplacian energy of threshold graphs let g be a connected threshold graph of order n with m edges and trace t in this paper we give a lower bound on laplacian energy in terms of n m and t of g from this we determine the threshold graphs with the first four minimal laplacian energies moreover we obtain the threshold graphs with the largest and the second largest laplacian energies

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

spectral properties of geometric arithmetic index the concept of geometric arithmetic index was introduced in the chemical graph theory recently but it has shown to be useful one of the main aims of algebraic graph theory is to determine how or whether properties of graphs are reflected in the algebraic properties of some matrices the aim of this paper is to study the geometric arithmetic index ga1 from an algebraic viewpoint since this index is related to the degree of the vertices of the graph our main tool will be an appropriate matrix that is a modification of the classical adjacency matrix involving the degrees of the vertices moreover using this matrix we define a ga laplacian matrix which determines the geometric arithmetic index of a graph and satisfies properties similar to the ones of the classical laplacian matrix

numerical methods for the solution of the third and fifth order dispersive korteweg de vries equations abstract two numerical methods are proposed for the solution of the third and fifth order korteweg de vries equations the first method is derived using central differences to replace the space derivatives with a predictor corrector time stepping and the second method by linearizing the implicit corrector scheme in which the solution is then found by solving a linear algebraic system at each time step rather than a nonlinear algebraic system which is more usual an important advantage to be gained from the use of the linearized implicit method over the predictor corrector method which is optimally stable is the ability to vary the mesh length the methods are analysed with respect to stability criteria and numerical dispersion numerical results portraying a single soliton solution and the interaction of more than one soliton are reported for the third order korteweg de vries equation numerical results for the fifth order korteweg de vries equation using the linearized implicit method are also reported

analytical and numerical aspects of certain nonlinear evolution equations i analytical abstract nonlinear partial difference equations are obtained which have as limiting forms the nonlinear schrodinger korteweg devries and modified korteweg devries equations these difference equations have a number of special properties they are constructed by methods related to the inverse scattering transform they can be used as a basis for numerical schemes to the associated nonlinear evolution equations experiments have shown that they compare very favorably with other known numerical methods papers ii iii in paper ii the ablowitz ladik scheme for the nonlinear schrodinger equation is compared to other known numerical schemes and generally proved to be faster than all utilized finite difference schemes but somewhat slower than the finite fourier pseudospectral methods in paper iii a proposed scheme for the korteweg de vries equation proved to be faster than both the finite difference and finite fourier methods already considered

information theoretic description of networks we present a new information theoretic approach for network characterizations it is developed to describe the general type of networks with n nodes and l directed and weighted links i e it also works for the simpler undirected and unweighted networks the new information theoretic measures for network characterizations are based on a transmitter receiver analogy of effluxes and influxes based on these measures we classify networks as either complex or non complex and as either democracy or dictatorship networks directed networks in particular are furthermore classified as either information spreading and information collecting networks the complexity classification is based on the information theoretic network complexity measure medium articulation ma it is proven that special networks with a medium number of links l n1 5 show the theoretical maximum complexity ma logn 2 2 a network is complex if its ma is larger than the average ma of appropriately randomized networks ma mar a network is of the democracy type if its redundancy r h b a it is an information collecting network otherwise an information spreading network finally different real networks directed and undirected weighted and unweighted are classified according to our general scheme

connective steiner 3 eccentricity index and network similarity measure abstract for a set s v g in a network g the steiner distance dg s of s is the minimum size among all connected subnetworks whose vertex sets contain s the steiner k eccentricity k v of a vertex v of g is the maximum steiner distance among all k vertex set s which contains the vertex v i e e k v max d s s v g s k v s based on steiner k eccentricity the connective steiner k eccentricity index is introduced as a newly structural invariant some properties of the connective steiner 3 eccentricity index are investigated firstly we present an o n2 polynomial time algorithm to calculate the connective steiner 3 eccentricity index of trees secondly some optimal problems among some network classes are discussed as its application finally we consider the network similarity measure based on the connective steiner 3 eccentricity index by two different methods we study its advantages numerical results show that the measure based on the connective steiner 3 eccentricity index has more advantages than the ones based on other topological indices graph energy randic index the largest adjacent eigenvalue the largest laplacian eigenvalue

adaptive multiquadric collocation for boundary layer problems an adaptive collocation method based upon radial basis functions is presented for the solution of singularly perturbed two point boundary value problems using a multiquadric integral formulation the second derivative of the solution is approximated by multiquadric radial basis functions this approach is combined with a coordinate stretching technique the required variable transformation is accomplished by a conformal mapping an iterated sine transformation a new error indicator function accurately captures the regions of the interval with insufficient resolution this indicator is used to adaptively add data centres and collocation points the method resolves extremely thin layers accurately with fairly few basis functions the proposed adaptive scheme is very robust and reaches high accuracy even when parameters in our coordinate stretching technique are not chosen optimally the effectiveness of our new method is demonstrated on two examples with boundary layers and one example featuring an interior layer it is shown in detail how the adaptive method refines the resolution

a rational spectral method for the kdv equation on the half line a petrov galerkin method using orthogonal rational functions is proposed for the korteweg de vries kdv equation on the half line with initial boundary values the nonlinear term and the right hand side term are treated by chebyshev rational interpolation explicitly and the linear terms are computed with the galerkin method implicitly such an approach is applicable using fast algorithms numerical results are presented for problems with both exponentially and algebraically decaying solutions respectively highlighting the performance of the proposed method

probabilistic inequalities for evaluating structural network measures proving interrelations between structural graph measures analytically has been intricate generally relations between structural graph measures describe the interplay between measures which turned out to be useful for better understanding the properties of such quantities the results which have been achieved so far are restricted to simple measures or special graph classes such as trees in this paper we introduce a probabilistic approach for establishing inequalities between quantitative network measures on random networks those inequalities between different graph measures lead to a deeper understanding of the mathematical apparatus and in particular to a discussion of quality aspects of structural graph measures which is a major contribution of this paper

the chebyshev legendre method implementing legendre methods on chebyshev points a new collocation method for the numerical solution of partial differential equations is presented this method uses the chebyshev collocation points but because of the way the boundary conditions are implemented it has all the advantages of the legendre methods in particular l 2 estimates can be easily obtained for hyperbolic and parabolic problems

on rainbow domination numbers of graphs given a graph g and a set of t colors assume that we assign an arbitrary subset of these colors to each vertex of g if we require that each vertex to which an empty set is assigned has in its neighborhood all t colors then this assignment is called a t rainbow dominating function of the graph g the corresponding invariant c r t g which is the minimum sum of numbers of assigned colors over all vertices of g is called the t rainbow domination number of g in this paper bounds for the t rainbow domination number of an arbitrary graph for an arbitrary positive integer t are given the 3 rainbow domination numbers of several classes of graphs such as paths cycles and the generalized petersen graphs p n k are investigated the 3 rainbow domination number of p n 1 is determined and the upper bounds for p n 2 and p n 3 are provided by computer search we obtain that the upper bounds for p n 2 match their exact values for n 17

graph distance measures based on topological indices revisited graph distance measures based on topological indices have been already explored by dehmer et al also inequalities for those graph distance measures have been proved in this paper we continue studying such comparative graph measures based on the well known wiener index graph energy and randic index respectively we prove extremal properties of the graph distance measures for some special classes of graphs to demonstrate useful properties of the measures we also discuss numerical results to conclude the paper we state some open problems

implicit explicit methods for time dependent partial differential equations implicit explicit imex schemes have been widely used especially in conjunction with spectral methods for the time integration of spatially discretized partial differential equations pdes of diffusion convection type typically an implicit scheme is used for the diffusion term and an explicit scheme is used for the convection term reaction diffusion problems can also be approximated in this manner in this work we systematically analyze the performance of such schemes propose improved new schemes and pay particular attention to their relative performance in the context of fast multigrid algorithms and of aliasing reduction for spectral methods for the prototype linear advection diffusion equation a stability analysis for first second third and fourth order multistep imex schemes is performed stable schemes permitting large time steps for a wide variety of problems and yielding appropriate decay of high frequency error modes are identified numerical experiments demonstrate that weak decay

a compact fourth order l stable scheme for reaction diffusion systems with nonsmooth data exponential time differencing runge kutta etdrk schemes based on diagonal pade approximations for the numerical solution of reaction diffusion systems containing nonsmooth data have the disadvantage of producing poor numerical results when the time steps are too large relative to the spatial steps on the other hand methods based on sub diagonal pade approximations do not suffer from this bottleneck in this paper we present a novel modified version of cox and matthews fourth order etdrk scheme based on sub diagonal pade approximation to matrix exponential functions in combination with compact fourth order finite difference scheme in space for direct integration of reaction diffusion problems with nonsmooth data for an efficient implementation of the scheme a partial fraction splitting technique is utilized in which it is required to solve several backward euler type linear systems at each time step moreover the design of the algorithm offers parallel implementation so we implement the proposed algorithm in parallel on two processors utilizing matlabmpi a parallel message passing version of matlab and obtain that the parallel version is computationally more efficient than the existing schemes considered in this paper we investigate the amplification factor of the scheme and plot its boundaries of stability regions which give an indication of the stability of the scheme calculation of the local truncation error and an empirical convergence analysis demonstrates the fourth order accuracy of the proposed scheme accuracy computational efficiency and reliability of the new scheme are demonstrated with numerical examples and comparing it with existing schemes

digraphs with large maximum wiener index recently the concept of wiener index was extended to digraphs which are not necessarily strongly connected and it was shown that some fundamental results extend naturally within this concept this extension could be applicable in the topics of directed large networks particularly because with this measure one assigns finite values to the average distance and betweenness centrality of the nodes in a directed network it is not hard to show that among digraphs on n vertices the directed cycle c n achieves the maximum wiener index next we investigate digraphs with the second maximum wiener index one can consider this problem in the realm of all digraphs or restricted to those obtained by directing undirected graphs so called antisymmetric digraphs in both situations we obtain that such digraphs are constructed from c n n 6 by adding a single arc we conclude the paper with consideration for possible further works

an inequality between the edge wiener index and the wiener index of a graph the wiener index w g of a connected graph g is defined to be the sum u vd u v of distances between all unordered pairs of vertices in g similarly the edge wiener index we g of g is defined to be the sum e fd e f of distances between all unordered pairs of edges in g or equivalently the wiener index of the line graph l g wu 2010 showed that we g w g for graphs of minimum degree 2 where equality holds only when g is a cycle similarly in knor et al 2014 it was shown that w e g 2 1 4 w g where denotes the minimum degree in g in this paper we extend improve these two results by showing that w e g 2 4 w g with equality satisfied only if g is a path on 3 vertices or a cycle besides this we also consider the upper bound for we g as well as the ratio w e g w g we show that among graphs g on n vertices w e g w g attains its minimum for the star

the energy of a graph abstract the energy e g of a simple graph g is defined to be the sum of the absolute values of the eigen values of g if g is a k regular graph on n vertices then e g k k n 1 n k b 2 and this bound is sharp it is shown that for each 0 there exist infinitely many n for each of which there exists a k regular graph g of order n with k e g b 2 two graphs with the same number of vertices are equienergetic if they have the same energy we show that for any positive integer n 3 there exist two equienergetic graphs of order 4n that are not cospectral

a legendre petrov galerkin and chebyshev collocation method for third order differential equations a legendre petrov galerkin lpg method for the third order differential equation is developed by choosing appropriate base functions the method can be implemented efficiently also this new approach enables us to derive an optimal rate of convergence in l2 norm the method is applied to some nonlinear problems such as the korteweg de vries kdv equation with the chebyshev collocation treatment for the nonlinear term it is a legendre petrov galerkin and chebyshev collocation lpg cc method numerical experiments are given to confirm the theoretical result

models of collective cell spreading with variable cell aspect ratio a motivation for degenerate diffusion models continuum diffusion models are often used to represent the collective motion of cell populations most previous studies have simply used linear diffusion to represent collective cell spreading while others found that degenerate nonlinear diffusion provides a better match to experimental cell density profiles in the cell modeling literature there is no guidance available with regard to which approach is more appropriate for representing the spreading of cell populations furthermore there is no knowledge of particular experimental measurements that can be made to distinguish between situations where these two models are appropriate here we provide a link between individual based and continuum models using a multi scale approach in which we analyze the collective motion of a population of interacting agents in a generalized lattice based exclusion process for round agents that occupy a single lattice site we find that the relevant continuum description of the system is a linear diffusion equation whereas for elongated rod shaped agents that occupy l adjacent lattice sites we find that the relevant continuum description is connected to the porous media equation pme the exponent in the nonlinear diffusivity function is related to the aspect ratio of the agents our work provides a physical connection between modeling collective cell spreading and the use of either the linear diffusion equation or the pme to represent cell density profiles results suggest that when using continuum models to represent cell population spreading we should take care to account for variations in the cell aspect ratio because different aspect ratios lead to different continuum models

a new bernoulli matrix method for solving second order linear partial differential equations with the convergence analysis in this paper a new matrix approach for solving second order linear partial differential equations pdes under given initial conditions has been proposed the basic idea includes integrating from the considered pdes and transforming them to the associated integro differential equations with partial derivatives therefore bernoulli operational matrices of differentiation and integration together with the completeness of bernoulli polynomials can be used for transforming integro differential equations to the corresponding algebraic equations a rigorous error analysis in the infinity norm is given provided that the known functions and the exact solution are sufficiently smooth and bounded a numerical example is included to demonstrate the validity and the applicability of the technique the results confirm the theoretical prediction

a history of graph entropy measures this survey seeks to describe methods for measuring the entropy of graphs and to demonstrate the wide applicability of entropy measures setting the scene with a review of classical measures for determining the structural information content of graphs we discuss graph entropy measures which play an important role in a variety of problem areas including biology chemistry and sociology in addition we examine relationships between selected entropy measures illustrating differences quantitatively with concrete examples

efficient spectral petrov galerkin methods for the integrated forms of third and fifth order elliptic differential equations using general parameters generalized jacobi polynomials abstract this article analyzes some algorithms for solving numerically the integrated forms of third and fifth order differential equations in one variable subject to homogenous and nonhomogeneous boundary conditions using a dual petrov galerkin method two new families of general parameters generalized jacobi polynomials are introduced and used for this purpose numerical results indicating the high accuracy and effectiveness of the proposed algorithms are presented

non monte carlo formulations and computational techniques for the stochastic non linear schrodinger equation stochastic ordinary and partial differential equations sopdes in various forms arise and are successfully utilized in the modeling of a variety of physical and engineered systems such as telecommunication systems electronic circuits cosmological systems financial systems meteorological and climate systems while the theory of stochastic partial and especially ordinary differential equations is more or less well understood there has been much less work on practical formulations and computational approaches to solving these equations in this paper we concentrate on the stochastic non linear schrodinger equation snlse that arises in the analysis of wave propagation phenomena mainly motivated by its predominant role as a modeling tool in the design of optically amplified long distance fiber telecommunication systems we present novel formulations and computational methods for the stochastic characterization of the solution of the snlse our formulations and techniques are not aimed at computing individual realizations i e sample paths for the solution of the snlse a la monte carlo instead starting with the snlse we derive new systems of differential equations and develop associated computational techniques the numerical solutions of these new equations directly produce the ensemble averaged stochastic characterization desired for the solution of the snlse in a non monte carlo manner without having to compute many realizations needed for ensemble averaging

efficient spectral galerkin method ii direct solvers of second and fourth order equations using chebyshev polynomials efficient direct solvers based on the chebyshev galerkin methods for second and fourth order equations are presented they are based on appropriate base functions for the galerkin formulation that lead to discrete systems with special structured matrices that can be efficiently inverted numerical results indicate that the direct solvers presented in this paper are significantly more accurate and efficient than those based on the chebyshev tau method

on the connective eccentricity index of trees and unicyclic graphs with given diameter abstract the connective eccentricity index cei of a graph g is defined as ce g v v g d v e v where e v d v denote the eccentricity and the degree of v respectively in this paper we investigate the connective eccentricity index of trees and unicyclic graphs on n vertices with diameter d we also characterize the extremal graphs

a direct discontinuous galerkin method for the generalized korteweg de vries equation energy conservation and boundary effect a direct discontinuous galerkin method for solving the generalized korteweg de vries kdv equation with both periodic data and non homogeneous boundary data is developed a class of numerical fluxes is identified so that the method preserves both momentum and energy for the initial value problem the method with such a property is numerically shown robust with less error in both phase and amplitude after long time simulation numerical examples are given to confirm the theoretical result and the capacity of this method for capturing soliton wave phenomena and various boundary wave patterns

hosoya entropy of fullerene graphs abstract entropy based methods are useful tools for investigating various problems in mathematical chemistry computational physics and pattern recognition in this paper we introduce a general framework for applying shannon entropy to fullerene graphs and used it to investigate their properties we show that important physical properties of these molecules can be determined by applying hosoya entropy to their corresponding graphs

a measure of similarity between graph vertices applications to synonym extraction and web searching we introduce a concept of similarity between vertices of directed graphs let ga and gb be two directed graphs with respectively na and nb vertices we define an nb times na similarity matrix s whose real entry sij expresses how similar vertex j in ga is to vertex i in gb we say that sij is their similarity score the similarity matrix can be obtained as the limit of the normalized even iterates of sk 1 bskat btska where a and b are adjacency matrices of the graphs and s0 is a matrix whose entries are all equal to 1 in the special case where ga gb g the matrix s is square and the score sij is the similarity score between the vertices i and j of g we point out that kleinberg s hub and authority method to identify web pages relevant to a given query can be viewed as a special case of our definition in the case where one of the graphs has two vertices and a unique directed edge between them in analogy to kleinberg we show that our similarity scores are given by the components of a dominant eigenvector of a nonnegative matrix potential applications of our similarity concept are numerous we illustrate an application for the automatic extraction of synonyms in a monolingual dictionary

the energy of graphs and matrices abstract given a complex m n matrix a we index its singular values as 1 a 2 a and call the value e a 1 a 2 a the energy of a thereby extending the concept of graph energy introduced by gutman let 2 m n a be an m n nonnegative matrix with maximum entry and a 1 n extending previous results of koolen and moulton for graphs we prove that e a a 1 m n m 1 a 2 2 a 1 2 m n n m m 2 furthermore if a is any nonconstant matrix then e a 1 a a 2 2 1 2 a 2 a finally we note that wigner s semicircle law implies that e g 4 3 o 1 n 3 2 for almost all graphs g

integration preconditioning of pseudospectral operators i basic linear operators this paper develops a family of preconditioners for pseudospectral approximations of pth order linear differential operators subject to various types of boundary conditions the approximations are based on ultraspherical polynomials with special attention being paid to legendre and chebyshev polynomial methods based on gauss lobatto quadrature points the eigenvalue spectrum of the preconditioned operators are obtained in closed analytic form and the weakly enforced boundary conditions are shown to result in a rank 2p perturbation of the identity operator i e the majority of the preconditioned eigenvalues are unity the spectrum of the preconditioned advective operator is shown to be bounded independent of the order of the approximation n however the preconditioned diffusive operator is in general indefinite with four real eigenvalues for dirichlet boundary conditions the spectral radius grows as sqrt n while it scales as n for the case of neumann boundary conditions these results are shown to be asymptotically optimal within the present framework generalizations to higher order differential operators general boundary conditions and arbitrary polynomial basis and quadrature nodes are discussed

fourier methods with extended stability intervals for the korteweg de vries equation a full leap frog fourier method for integrating the korteweg de vries kdv equation u t uu x varepsilon u xxx 0 results in an o n 3 stability constraint on the time step where n is the number of fourier modes used this stability limit is much more restrictive than the accuracy limit for many applications in this paper we propose a method for which the stability limit is extended by treating the linear dispersive u xxx term implicitly thus timesteps can be taken up to an accuracy limit larger than the explicit stability limit the implicit method is implemented without solving linear systems by integrating in time in the fourier space and discretizing the nonlinear uu x term by leap frog a second method we propose uses basis functions which solve the linear part of the kdv equation and leap frog for time integration a linearized stability analysis of the proposed schemes proves that a version of the first scheme possesses a certain kind of unconditional stability and that

a new dual petrov galerkin method for third and higher odd order differential equations application to the kdv equation a new dual petrov galerkin method is proposed analyzed and implemented for third and higher odd order equations using a spectral discretization the key idea is to use trial functions satisfying the underlying boundary conditions of the differential equations and test functions satisfying the dual boundary conditions the method leads to linear systems which are sparse for problems with constant coefficients and well conditioned for problems with variable coefficients our theoretical analysis and numerical results indicate that the proposed method is extremely accurate and efficient and most suitable for the study of complex dynamics of higher odd order equations

a pseudospectral procedure for the solution of nonlinear wave equations with examples from free surface flows an algorithm for the solution of general isotropic nonlinear wave equations is presented the algorithm is based on a symmetric factorization of the linear part of the wave operator followed by its exact integration through an integrating factor in spectral space the remaining nonlinear and forcing terms can be handled with any standard pseudospectral procedure solving the linear part of the wave operator exactly effectively eliminates the stiffness of the original problem characterized by a wide range of temporal scales the algorithm is tested and applied to several problems of three dimensional long surface waves solitary wave propagation interaction diffraction and the generation of waves by flow over slowly varying bottom topography other potential applications include waves in rotating and stratified flows and wave interaction with more pronounced topographic features

comparing large scale graphs based on quantum probability theory abstract in this paper a new measurement to compare two large scale graphs based on the theory of quantum probability is proposed an explicit form for the spectral distribution of the corresponding adjacency matrix of a graph is established our proposed distance between two graphs is defined as the distance between the corresponding moment matrices of their spectral distributions it is shown that the spectral distributions of their adjacency matrices in a vector state includes information not only about their eigenvalues but also about the corresponding eigenvectors moreover we prove that the proposed distance is graph invariant and sub structure invariant examples with various graphs are given and distances between graphs with few vertices are checked computational results for real large scale graphs show that its accuracy is better than any existing methods and time cost is extensively cheap

analytical and numerical aspects of certain nonlinear evolution equations 1v numerical modified korteweg de vries equation various numerical methods are used in order to approximate the korteweg de vries equation namely i zabusky kruskal scheme ii hopscotch method iii a scheme due to goda iv a proposed local scheme v a proposed global scheme vi a scheme suggested by kruskal vii split step fourier method by tappert viii an improved split step fourier method and ix pseudospectral method by fornberg and whitham comparisons between our proposed scheme which is developed using notions of the inverse scattering transform and the other utilized schemes are obtained

on ordering of complements of graphs with respect to matching numbers we consider an earlier much studied quasi order defined in terms of matching numbers of graphs and apply it to graph complements we establish four transformations on the complements of graphs that increase or decrease the matching numbers accordingly several applications of these results are put forward

applications of a theorem by ky fan in the theory of graph energy abstract the energy of a graph g is equal to the sum of the absolute values of the eigenvalues of g which in turn is equal to the sum of the singular values of the adjacency matrix of g let x y and z be matrices such that x y z the ky fan theorem establishes an inequality between the sum of the singular values of z and the sum of the sum of the singular values of x and y this theorem is applied in the theory of graph energy resulting in several new inequalities as well as new proofs of some earlier known inequalities

numerical methods for the generalized zakharov system we present two numerical methods for the approximation of the generalized zakharov system zs the first one is the time splitting spectral tssp method which is explicit time reversible and time transverse in variant if the generalized zs is keeps the same decay rate of the wave energy as that in the generalized zs gives exact results for the plane wave solution and is of spectral order accuracy in space and second order accuracy in time the second one is to use a local spectral method the discrete singular convolution dsc for spatial derivatives and the fourth order runge kutta rk4 for time integration which is of high the same as spectral order accuracy in space and can be applied to deal with general boundary conditions in order to test accuracy and stability we compare these two methods with other existing methods fourier pseudospectral method fps and wavelet galerkin method wg for spatial derivatives combining with the rk4 for time integration as well as the standard finite difference method fd for solving the zs with a solitary wave solution furthermore extensive numerical tests are presented for plane waves solitary wave collisions in 1d as well as a 2d problem of the generalized zs numerical results show that tssp and dsc are spectralorder accuracy in space and much more accurate than fd and for stability tssp requires k o h dsc rk4 requires k o h2 for fixed acoustic speed where k is the time step and h is the spatial mesh size

numerical solution of stochastic fractional integro differential equation by the spectral collocation method fractional calculus is used to model various different phenomena in nature today the aim of this paper is to propose the shifted legendre spectral collocation method to solve stochastic fractional integro differential equations sfides in this approach we consider the p panels m point newtoncotes rules with m fixed for estimating it integrals the main characteristic of the presented method is that it reduces sfides into a system of algebraic equations thus we can solve the problem by newtons method furthermore the convergence analysis of the approach is considered the method is computationally attractive and numerical examples confirm the validity and efficiency of the proposed method