on the sum of laplacian eigenvalues of a signed graph abstract for a signed graph let e denote the number of edges and s k denote the sum of the k largest eigenvalues of the laplacian matrix of we conjecture that for any signed graph with n vertices s k e k 1 2 1 holds for k 1 n we prove the conjecture for any signed graph when k 2 and prove that this conjecture is true for unicyclic and bicyclic signed graphs

lower bounds for the least laplacian eigenvalue of unbalanced blocks abstract we denote by n and n the number of vertices and the least laplacian eigenvalue of a signed graph respectively a connected unbalanced signed graph without cut vertices is called an unbalanced block we prove that n g 4 l u n holds for every unbalanced block g where l u denotes the length of the longest negative cycle in g we also prove that n g k 2 12 g n 1 n 2 n k 2 2 12 g n 1 n 2 n k 4 g n 1 n 2 n k being the geometric mean of given arguments holds for every signed graph g which contains k edge disjoint spanning subgraphs such that the least laplacian eigenvalue of the ith of them is not less than the least laplacian eigenvalue of the negative cycle c n i using this result we prove that n g k 2 12 n 2 2 12 n 4 holds for every unbalanced block with k edge disjoint negative hamiltonian cycles

a mathematical model of adomian polynomials the adomian decomposition method is reviewed and a mathematical model of adomian polynomials is introduced to show the convenience of this model for computer programming the mathematica package is used to generate the adomian polynomials for any order

convergent series solution of nonlinear equations abstract the author s decomposition method 1 provides a new efficient computational procedure for solving large classes of nonlinear and or stochastic equations these include differential equations containing polynomial exponential and trigonometric terms negative or irrational powers and product nonlinearities 2 also included are partial differential equations 3 delay differential equations 4 algebraic equations 5 and matrix equations 6 which describe physical systems essentially the method provides a systematic computational procedure for equations containing any nonlinear terms of physical significance the procedure depends on calculation of the author s a n a finite set of polynomials 1 13 in terms of which the nonlinearities can be expressed this paper shows important properties of the a n which ensure an accurate and computable convergent solution by the author s decomposition method 1 since the nonlinearities and or stochasticity which can be handled are quite general the results are potentially extremely useful for applications and make a number of common approximations such as linearization unnecessary

a new approach to the wave equation an application of the decomposition method abstract in the present paper the inhomogeneous wave equation is solved quickly easily and elegantly by adomian s decomposition method

signed bicyclic graphs minimizing the least laplacian eigenvalue abstract a signed graph is a pair g where g v g e g is a graph and e g 1 1 is the sign function on the edges of g for a signed graph we consider the laplacian matrix defined as l d g a where d g is the matrix of vertices degrees of g and a is the signed adjacency matrix the least laplacian eigenvalue is zero if and only if the signed graph is balanced i e all cycles contain an even number of negative edges here we show that among the unbalanced bicyclic signed graphs of given order n 5 the least laplacian eigenvalue is minimal for signed graphs consisting of two triangles only one of which is unbalanced connected by a path we also identify the signed graphs minimizing the least eigenvalue among those whose unbalanced bicyclic base is a theta graph

signless laplacians of finite graphs we survey properties of spectra of signless laplacians of graphs and discuss possibilities for developing a spectral theory of graphs based on this matrix for regular graphs the whole existing theory of spectra of the adjacency matrix and of the laplacian matrix transfers directly to the signless laplacian and so we consider arbitrary graphs with special emphasis on the non regular case the results which we survey old and new are of two types a results obtained by applying to the signless laplacian the same reasoning as for corresponding results concerning the adjacency matrix b results obtained indirectly via line graphs among other things we present eigenvalue bounds for several graph invariants an interpretation of the coefficients of the characteristic polynomial a theorem on powers of the signless laplacian and some remarks on star complements

the decomposition method applied to chandrasekhar h equation in this paper adomian s decomposition method is applied to the nonlinear chandrasekhar h equation the solution is expressed as an infinite series it is shown that two or three iterations are sufficient to obtain good approximation of the exact solution adomian s method has been illustrated by considering some particular cases of this nonlinear integral equation that are of physical interest

a new approach to the heat equation an application of the decomposition method abstract the author s decomposition method adomian stochastic systems academic press new york 1983 nonlinear stochastic operator equations academic press new york in press bellman and adomian partial differential equations methods for their treatment and solution reidel dordrecht 1984 adomian j comput appl math 2 1984 225 230 shown elsewhere to provide computable accurate approximate convergent solutions to nonlinear stochastic operator equations is applied to the heat equation

the decomposition approach to inverse heat conduction adomian s decomposition approach is employed for solving some inverse boundary value problems in heat conduction further the mollification method is applied to deal with noisy input data and obtain a stable approximate solution

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

a new approach to nonlinear partial differential equations abstract the author s decomposition method for the solution of operator equations which may be nonlinear and or stochastic is generalized to multidimensional cases

construction of n ary f polygroups by an n ary f hyperoperation on a non empty set a we mean a function from a a to the set of all non zero fuzzy subsets of a where a appears n times using this concept we introduce the notion of n ary f polygroups which is a generalization of ideas presented by zahedi hasankhani and davvaz et al some basic results are obtained

a computational algebraic investigation of the decomposition method for time dependent problems the convergence of the decomposition method as applied to time dependent problems governed by the heat or wave equations is investigated numerically using computational algebraic techniques

signed graphs with extremal least laplacian eigenvalue abstract a signed graph is a pair g where g v g e g is a graph and e g is the corresponding sign function for a signed graph we consider the laplacian matrix defined as l d g a where d g is the matrix of vertex degrees of g and a is the signed adjacency matrix it is well known that is balanced that is each cycle contains an even number of negative edges if and only if the least laplacian eigenvalue n 0 therefore if is not balanced then n 0 we show here that among unbalanced connected signed graphs of given order the least eigenvalue is minimal for an unbalanced triangle with a hanging path while the least eigenvalue is maximal for the complete graph with the all negative sign function

balancedness and the least eigenvalue of laplacian of signed graphs abstract let g be a connected signed graph where g is the underlying simple graph and e g 1 1 is the sign function on the edges of g let l d g a be the laplacian of and 1 2 n 0 be its eigenvalues it is well known that a signed graph is balanced if and only if n 0 here we show that if is unbalanced then n estimates how much is far from being balanced in particular let resp be the frustration number resp frustration index that is the minimum number of vertices resp edges to be deleted such that the signed graph is balanced then we prove that n further we analyze the case when n in the latter setting we identify the structure of the underlying graph g and we give an algebraic condition for l which leads to the above equality

sinc galerkin method for solving nonlinear boundary value problems the sinc galerkin method is used to approximate solutions of nonlinear problems involving nonlinear second fourth and sixth order differential equations with homogeneous and nonhomogeneous boundary conditions the scheme is tested on four nonlinear problems the results demonstrate the reliability and efficiency of the algorithm developed

the decomposition method for forward and backward time dependent problems the convergence of the decomposition method as applied to time dependent problems governed by the heat wave and beam equations is investigated for both forward direct and backward inverse problems it is shown that for forward problems the convergence is faster than for backward problems

a sharp lower bound for the least eigenvalue of the signless laplacian of a non bipartite graph we prove that the minimum value of the least eigenvalue of the signless laplacian of a connected non bipartite graph with a prescribed number of vertices is attained solely in the unicyclic graph obtained from a triangle by attaching a path at one of its endvertices

on product nonlinearities in stochastic differential equations the author s methods for solution of stochastic differential equations shown elsewhere to be applicable for nonlinear cases including polynomial exponential and trigonometric terms is shown to be applicable to product nonlinearities as well

application of the adomian decomposition method for two dimensional parabolic equation subject to nonstandard boundary specifications in this work we are concerned with the solution of a nonlocal boundary value problem an approach is presented for solving the two dimensional parabolic partial differential equation subject to integral boundary specifications the main objective is to propose an alternative method of solution one not based on finite difference methods or finite element schemes or spectral techniques the aim of the present paper is to investigate the application of the adomian decomposition method for solving the two dimensional linear parabolic partial differential equation with nonlocal boundary specifications replacing the classical boundary conditions the adomian decomposition method is used by many researchers to investigate several scientific applications and requires less work if compare with the traditional techniques the introduction of this idea as will be discussed not only provides the solution in a series form but it also guarantees considerable saving of the calculations volume the solutions will be handle more easily quickly and elegantly without linearizing the problem by implementing the decomposition method rather than the standard methods for the exact solutions in this approach the solution is found in the form of a convergent power series with easily computed components the adomian decomposition scheme is easy to program in applied problems and provides immediate and convergent solutions without any need for linearization or discretization to give a clear overview of the methodology we have selected illustrative example

on signed graphs with just two distinct laplacian eigenvalues abstract we investigate signed graph whose laplacian matrix has just two distinct eigenvalues all simple connected signed graphs with maximum degree at most 4 and just two distinct laplacian eigenvalues are characterized in this paper

f polygroups ii first a product between two f subpolygroups of a given f polygroup is defined and then a necessary and sufficient condition is given in order that the product be an f subpolygroup then by considering the notion of homomorphism of f polygroups some related results are obtained finally the concept of quotient f polygroup is defined and then some theorems are proved

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

some remarks on congruences obtained from the l fuzzy nakano hyperoperation in this paper we study relations which are congruences with respect to and p where p is the p cut of the l fuzzy hyperoperation the main idea is to start from an equivalence relation r 1 which is a congruence with respect to and 1 and for each email protected x construct an equivalence relation r p which is a congruence with respect to and p furthermore for each email protected r p we construct a quotient hyperoperation p and we show that the hyperalgebra x r p p is a join space and the hyperalgebra x r p p p is a hyperlattice

the decomposition method for approximate solution of the goursat problem abstract in this paper the adomian s decomposition method is effectively implemented to establish an analytic solution and a reliable numerical approximation to the goursat partial differential equation linear and nonlinear goursat examples are examined and the method supplied quantitatively reliable results for these types of problems the accuracy level of the results obtained indicates the superiority of the decomposition method over existing numerical methods that were applied to the goursat problem

a technique for approximate solutions to schrodinger like equations in the present paper the schro dinger like equations are solved easily quickly and elegantly using adomian s decomposition method the technique has been illustrated by some examples which are of physical interest

the numerical solution of fifth order boundary value problemsby the decomposition method in this paper we present a fast and accurate numerical scheme for the solution of fifth order boundary value problems with two point boundary conditions the adomian decomposition method and a modified form of this method are applied to construct the numerical solution the new approach provides the solution in the form of a rapidly convergent series and not at grid points two numerical illustrations are given to show the pertinent features of technique

inversion of nonlinear stochastic operators abstract the operator theoretic method adomian and malakian j math anal appl 76 1 1980 183 201 recently extended adomian s solutions of nonlinear stochastic differential equations g adomian stochastic systems analysis in applied stochastic processes nonlinear stochastic differential equations j math anal appl 55 1 1976 441 452 on the modeling and analysis of nonlinear stochastic systems in proceeding international conf on mathematical modeling vol 1 pp 29 40 to provide an efficient computational procedure for differential equations containing polynomial exponential and trigonometric nonlinear terms n y the procedure depends on the calculation of certain quantities an and bn this paper generalizes the calculation of the an and bn to much wider classes of nonlinearities of the form n y y essentially the method provides a systematic computational procedure for differential equations containing any nonlinear terms of physical significance this procedure depends on a recurrence rule from which explicit general formulae are obtained for the quantities an and bn for any order n in a convenient form this paper also demonstrates the significance of the iterative series decomposition proposed by adomian for linear stochastic operators in 1964 and developed since 1976 for nonlinear stochastic operators since both the nonlinear and stochastic behavior is quite general the results are extremely significant for applications processes need not for example be limited to either gaussian processes white noise or small fluctuations

the l fuzzy nakano hypergroup in this paper we start with a lattice x and define in terms of a family of crisp hyperoperations p one hyperoperation for each p x we show that for every p the hyperalgebra x p is a join space and the hyperalgebra x p is very similar to a hyperlattice then we use the hyperoperations p as p cuts to introduce an l fuzzy hyperoperation and show that x is an l fuzzy join space and the hyperalgebra x p is very similar to an l fuzzy hyperlattice

a learning approach to the two person decentralized team problem with incomplete information a learning approach to the two person decentralized team problem with incomplete information and with a 2x2 payoff matrix is considered it is shown that if the payoff matrix is unimodal there exists a proper choice of parameters of the learning algorithm that will ensure asymptotically an expected payoff as close to maximum payoff as desired it is shown that a team problem with multimodal payoff matrix gives rise to interesting class of open problems

fuzzy hyperrings hv rings based on fuzzy universal sets the aim of this paper is to define and study the concept of a fuzzy hyperring which depends on the concept of a fuzzy universal set a correspondence relation between a fuzzy hyperring based on a fuzzy universal set and both ordinary and fuzzy hyperideal in the sense of davvaz is obtained