2 walk regular graphs with a small number of vertices compared to the valency abstract in 2013 it was shown that for a given real number 2 there are only finitely many distance regular graphs with valency k and diameter d 3 having at most k vertices except for the following two cases i d 3 and is imprimitive i i d 4 and is antipodal and bipartite in this paper we will generalize this result to 2 walk regular graphs in this case also incidence graphs of certain group divisible designs appear

a characterization of bipartite distance regular graphs abstract it is well known that the halved graphs of a bipartite distance regular graph are distance regular examples are given to show that the converse does not hold thus a natural question is to find out when the converse is true in this paper we give a quasi spectral characterization of a connected bipartite weighted 2 punctually distance regular graph whose halved graphs are distance regular in the case the spectral diameter is even we show that the graph characterized above is distance regular

2d parallel and stable group explicit finite difference method for solution of diffusion equation abstract recently various versions of alternating group explicit or alternating group explicit implicit methods were proposed for solution of diffusion equation the main benefits of these methods are good stability accuracy and parallelizing but these methods were developed for 1d case and stability and accuracy were investigated for 1d case too in the present study we extend the new group explicit method r tavakoli p davami new stable group explicit finite difference method for solution of diffusion equation appl math comput 181 2006 1379 1386 to 2d with operator splitting method the implementation of method is discussed in details our numerical experiment shows that such 2d extension is unconditionally stable and it is more accurate that traditional unconditional stable explicit method

regular graphs with four eigenvalues we study the connected regular graphs with four distinct eigenvalues properties and feasibility conditions of the eigenvalues are found several examples constructions and characterizations are given as well as some uniqueness and nonexistence results

structural stability for variable exponent elliptic problems ii the p u laplacian and coupled problems abstract we study well posedness for elliptic problems under the form b u div a x u u f where a satisfies the classical leray lions assumptions with an exponent p that may depend both on the space variable x and on the unknown solution u a prototype case is the equation u div u p u 2 u f we have to assume that inf x z r p x z is greater than the space dimension n then under mild regularity assumptions on and on the nonlinearities we show that the associated solution operator is an order preserving contraction in l 1 in addition existence analysis for a sample coupled system for unknowns u v involving the p v laplacian of u is carried out coupled elliptic systems with similar structure appear in applications e g in modelling of stationary thermorheological fluids

distance regular graphs with small number of distinct distance eigenvalues abstract in this paper we characterize distance regular graphs with diameter three having exactly three distinct distance eigenvalues and also bipartite distance regular graphs with diameter four having three distinct distance eigenvalues we derive some properties and give particular examples of such graphs we also present an infinite family of bipartite semiregular graphs with diameter four having exactly four distinct distance eigenvalues with these results we address some problems posed in atik and panigrahi 2015 3

unconditional stability of parallel alternating difference schemes for semilinear parabolic systems the general alternating schemes with intrinsic parallelism for semilinear parabolic systems are studied first we prove the a priori estimates in the discrete h 1 space of the difference solution for these schemes then the existence of the difference solution for these schemes follows from the fixed point principle finally the unconditional stability of the general alternating schemes is proved the alternating group explicit scheme the alternating segment explicit implicit scheme and the alternating segment crank nicolson scheme are the special cases of the general alternating schemes

interlacing eigenvalues and graphs abstract we give several old and some new applications of eigenvalue interlacing to matrices associated to graphs bounds are obtained for characteristic numbers of graphs such as the size of a maximal co clique the chromatic number the diameter and the bandwidth in terms of the eigenvalues of the standard adjacency matrix or the laplacian matrix we also deal with inequalities and regularity results concerning the structure of graphs and block designs

geometric aspects of 2 walk regular graphs abstract a t walk regular graph is a graph for which the number of walks of given length between two vertices depends only on the distance between these two vertices as long as this distance is at most t such graphs generalize distance regular graphs and t arc transitive graphs in this paper we will focus on 1 and in particular 2 walk regular graphs and study analogues of certain results that are important for distance regular graphs we will generalize delsarte s clique bound to 1 walk regular graphs godsil s multiplicity bound and terwilliger s analysis of the local structure to 2 walk regular graphs we will show that 2 walk regular graphs have a much richer combinatorial structure than 1 walk regular graphs for example by proving that there are finitely many non geometric 2 walk regular graphs with given smallest eigenvalue and given diameter a geometric graph is the point graph of a special partial linear space a result that is analogous to a result on distance regular graphs such a result does not hold for 1 walk regular graphs as our construction methods will show

the d yakonov fully explicit variant of the iterative decomposition method abstract in this paper a new iterative alternating decomposition iade scheme of 4 2 order of accuracy is developed to solve the one dimensional parabolic problem it is based on the two stage fractional splitting strategy suggested by d yakonov and found to be generally more accurate than the recently developed 2 2 accurate alternating group explicit age method of peaceman rachford variant as the method is fully explicit its feature can be fully utilized for parallelization by means of a domain decomposition strategy

there are only finitely many distance regular graphs of fixed valency greater than two in this paper we prove the bannai ito conjecture namely that there are only finitely many distance regular graphs of fixed valency greater than two

a non separable solution of the diffusion equation based on the galerkin s method using cubic splines abstract the two dimensional diffusion equation of the form 2 u x 2 2 u y 2 1 d u t is considered in this paper we try a bi cubic spline function of the form i j 0 n n c i j t b i x b j y as its solution the initial coefficients c i j 0 are computed simply by applying a collocation method c i j f x i y j where f x y u x y 0 is the given initial condition then the coefficients c i j t are computed by x t e tq x 0 where x t c 0 1 c 0 1 c 0 2 c 0 n c 1 0 c n n is a one dimensional array and the square matrix q is derived from applying the galerkin s method to the diffusion equation note that this expression provides a solution that is not necessarily separable in space coordinates x y the results of sample calculations for a few example problems along with the calculation results of approximation errors for a problem with known analytical solution are included

some results on the p u laplacian problem the p laplacian problem with the exponent of nonlinearity p depending on the solution u itself is considered in this work both situations when p u is a local quantity or when p u is nonlocal are studied here for the associated boundary value local problem we prove the existence of weak solutions by using a singular perturbation technique we also prove the existence of weak solutions to the nonlocal version of the associated boundary value problem the issue of uniqueness for these problems is addressed in this work as well in particular by working out the uniqueness for a one dimensional local problem and by showing that the uniqueness is easily lost in the nonlocal problem

the infinitude of 7 arc transitive graphs abstract by theorems of tutte weiss and others it is known that there are no finite symmetric graphs of degree greater than 2 with automorphism group transitive on 8 arcs and that 7 arc transitivity can occur only in the case of graphs of degree 3 m 1 in this article it is shown that there are infinitely many 7 arc transitive finite quartic graphs indeed for all but finitely many positive integers n there is a finite connected 7 arc transitive quartic graph with the alternating group a n acting transitively on its 7 arcs and another with the symmetric group s n acting transitively on its 7 arcs the proof uses a construction involving permutation representations of a generic infinite group to produce an infinite family of finite graphs with the required properties

new stable group explicit finite difference method for solution of diffusion equation a new group explicit method for solution of diffusion equation is presented this method is based on domain decomposition concept and using asymmetric saul yev schemes for internal nodes of each sub domain and alternating group explicit method for sub domain s interfacial nodes this new method has several advantages such as good parallelism unconditional stability fully explicit nature and better accuracy than original saul yev schemes the details of implementation and proving stability are briefly discussed numerical experiments on stability and accuracy are also presented

an unconditionally stable parallel difference scheme for parabolic equations an unconditionally stable difference scheme with intrinsic parallelism is constructed in this paper taking the parabolic equation u t u x x as an example the main ideas are described as follows we decompose the domain w into some overlapping subdomains take values of the last time layer as values of the time layer on inner boundary points of subdomains solve it with the fully implicit scheme inside each subdomain and then take correspondent values of its neighbor subdomains as their values for inner boundary points of each subdomain and mean of its neighbor subdomain and itself at overlapping points its unconditional stability and convergence are proved though its truncation error is o 1 we will still prove that the estimate order is improved to o t h

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

characterizing l m walk regular graphs abstract a graph g with diameter d and d 1 distinct eigenvalues is said to be l m walk regular for some integers l 0 d and m 0 d l m if the number of walks of length i 0 l between any pair of vertices at distance j 0 m depends only on the values of i and j in this paper we study some algebraic and combinatorial characterizations of l m walk regularity based on the so called predistance polynomials and the preintersection numbers

implicit explicit parallel asynchronous solver of parabolic pdes this paper presents a method for the solution of parabolic pdes on parallel computers which is a combination of implicit and explicit finite difference schemes based on a domain decomposition dd strategy moreover this method is asynchronous i e no explicit synchronization is required among processors we determine the values at subdomains boundaries by our new high order asynchronous explicit schemes then any known high order implicit finite difference scheme can be applied within each subdomain we present a technique for derivation of appropriate asynchronous explicit schemes based on green s functions synchronous versions of these schemes are obtained as special cases the applicability of this method is also demonstrated for a family of nonlinear problems our new explicit schemes are of high order and yet stable for a large time step as established in our analysis of their numerical properties moreover these schemes provide attractive properties for parallel implementation being asynchronous they allow local time stepping thus eliminating the need for a global synchronized time step moreover our asynchronous computation is time stabilizing in the sense that the calculation implicitly prevents a growing time gap between neighboring subdomains the locality property due to the exponential decay of green s functions implies that communication is needed only between neighboring processors hence this method which is designed to minimize the overhead associated with the synchronization of the multiple processors is specifically suitable for parallel computers having a high synchronization cost or highly varying load even in cases in which some processors have persistent speed differences furthermore the implementation of different resolution in each subdomain e g irregular or unstructured grid makes it valuable as an adaptive algorithm the above schemes were implemented and tested on the shared memory multi user cray j90 and sequent balance machines these implementations prove high accuracy and high degree of parallelism this work is complementary to our previous work on asynchronous schemes comput math appl 24 1992 pp 33 53 appl numer math 12 1993 pp 27 45 numer algorithms 6 1994 pp 275 296 numer algorithms 12 1996 pp 159 192

heatlet approach to diffusion equation on unbounded domains we develop heatlets the fundamental solutions of heat equation using wavelets for numerically solving inhomogeneous and homogeneous initial value problems of diffusion equation on unbounded domains unlike finite difference and finite element methods diffusion into an infinite medium is satisfied analytically avoiding the need for artificial boundary conditions on a finite computational domain the approach is applied to a number of examples and the numerical results confirm the theoretical findings

on the numerical solution of the diffusion equation with variable space operator in present paper numerical schemes are developed for obtaining approximate solutions to the mixed problem for one dimensional diffusion equation with variable space operator the method of lines semidiscretization approach is used to transform the model partial differential equation into a system of first order linear ordinary differential equations the partial derivative with respect to the space variable is approximated by the first and second order finite difference approximation for the solution of the resulting system of first order ordinary differential equations we apply the first and second order of accuracy difference schemes stability estimates for the solution of these difference schemes are established numerical techniques are developed by applying a procedure of the solution of first order linear difference equation with matrix coefficients the algorithms are tested on a model problem in biofluid mechanics two regions are considered close to the endothelial cells ec which can be modeled taking the mixed problem for one dimensional diffusion equation with variable space operator

efficient numerical schemes for gradient vector flow since its publication more than 10 years ago the gradient vector flow gvf technique has been used and adapted to various models and problems its effectiveness has greatly contributed to its popularity the main drawback of gvf and its generalisation however is their expensive computation load and its consequence on the capture range in this work we propose and compare different efficient numerical schemes to solve the gvf and its generalisations

b spline galerkin methods for numerical solutions of the burgers equation the first order splitting method is applied to the burgers equation and two numerical algorithms are set up for this time splitted burgers equation using both quadratic and cubic b spline galerkin finite element techniques the unknowns in the equation is resolved with the b splines in space and standart crank nicolson differencing schemes is used to advance solution in time results of three numerical test problems are used to compare each schemes with available some results

an inequality involving the second largest and smallest eigenvalue of a distance regular graph abstract for a distance regular graph with second largest eigenvalue resp smallest eigenvalue 1 resp d we show that 1 1 d 1 b 1 holds where equality only holds when the diameter equals two using this inequality we study distance regular graphs with fixed second largest eigenvalue

on perturbations of almost distance regular graphs in this paper we show that certain almost distance regular graphs the so called h punctually walk regular graphs can be characterized through the cospectrality of their perturbed graphs a graph g with diameter d is called h punctually walk regular for a given hd if the number of paths of length between a pair of vertices u v at distance h depends only on the graph perturbations considered here are deleting a vertex adding a loop adding a pendant edge adding removing an edge amalgamating vertices and adding a bridging vertex we show that for walk regular graphs some of these operations are equivalent in the sense that one perturbation produces cospectral graphs if and only if the others do our study is based on the theory of graph perturbations developed by cvetkovic godsil mckay rowlinson schwenk and others as a consequence some new characterizations of distance regular graphs are obtained