a review of algebraic multigrid abstract since the early 1990s there has been a strongly increasing demand for more efficient methods to solve large sparse unstructured linear systems of equations for practically relevant problem sizes classical one level methods had already reached their limits and new hierarchical algorithms had to be developed in order to allow an efficient solution of even larger problems this paper gives a review of the first hierarchical and purely matrix based approach algebraic multigrid amg amg can directly be applied for instance to efficiently solve various types of elliptic partial differential equations discretized on unstructured meshes both in 2d and 3d since amg does not make use of any geometric information it is a plug in solver which can even be applied to problems without any geometric background provided that the underlying matrix has certain properties

a note on the positive semidefiniteness of aα g abstract let g be a graph with adjacency matrix a g and let d g be the diagonal matrix of the degrees of g for every real 0 1 write a g for the matrix a g d g 1 a g let 0 g be the smallest for which a g is positive semidefinite it is known that 0 g 1 2 the main results of this paper are 1 if g is d regular then 0 min a g d min a g where min a g is the smallest eigenvalue of a g 2 g contains a bipartite component if and only if 0 g 1 2 3 if g is r colorable then 0 g 1 r

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

bvφ solutions of nonlinear integral equations abstract in this paper we investigate solutions of nonlinear hammerstein and volterra hammerstein integral equations in the space of functions of bounded variation in the sense of young we prove the existence and in some cases the existence and uniqueness of local and global solutions in this class real valued as well as vector valued functions are under our consideration the method of our proofs is based on an application of the banach contraction principle as well as the leray schauder alternative for contractions

local fourier analysis of multigrid for the curl curl equation we present a local fourier analysis of multigrid methods for the two dimensional curl curl formulation of maxwell s equations both the hybrid smoother proposed by hiptmair and the overlapping block smoother proposed by arnold falk and winther are considered the key to our approach is the identification of two dimensional eigenspaces of the discrete curl curl problem by decoupling the fourier modes for edges with different orientations this procedure is used to quantify the smoothing properties of the considered smoothers and the convergence behavior of the multigrid methods additionally we identify the helmholtz splitting in fourier space this allows several well known properties to be recovered in fourier space such as the commutation properties of the classical nedelec prolongator and the equivalence of the curl curl operator and the vector laplacian for divergence free vectors we show how the approach used in this paper can be generalized to two and three dimensional problems in h curl and h div and to other types of regular meshes

on the multiplicity of an arbitrary aα eigenvalue of a connected graph abstract the a matrix of a graph g is defined by nikiforov as a g d g 1 a g where 0 1 a g and d g respectively denotes the adjacency matrix and the degree diagonal matrix of g the eigenvalues of a g are called the a eigenvalues of g in this paper we prove that for a connected graph g of order n and with maximum degree 2 and for any 0 1 the multiplicity of an arbitrary a eigenvalue of g which is denoted as m g is bounded by m g 2 n 2 1 the equality holds if and only if g and satisfy one of the following conditions i g k n with n 3 and n 1 ii g c n with even order and 2 2 1 cos 2 j n j 1 2 n 2 2 iii g c n with odd order and 2 2 1 cos 2 j n j 1 2 n 1 2 iv g k n 2 n 2 with n 4 and n 2 applying this result we solve an open conjecture posed by zhou et al in 32 one year ago

on the aα characteristic polynomial of a graph abstract let g be a graph with n vertices and let a g and d g denote respectively the adjacency matrix and the degree matrix of g define a g d g 1 a g for any real 0 1 the a characteristic polynomial of g is defined to be det x i n a g j c j g x n j where det denotes the determinant of and i n is the identity matrix of size n the a spectrum of g consists of all roots of the a characteristic polynomial of g a graph g is said to be determined by its a spectrum if all graphs having the same a spectrum as g are isomorphic to g in this paper we first formulate the first four coefficients c 0 g c 1 g c 2 g and c 3 g of the a characteristic polynomial of g and then we observe that a spectra are much efficient for us to distinguish graphs by enumerating the a characteristic polynomials for all graphs on at most 10 vertices to verify this observation we characterize some graphs determined by their a spectra

fourier analysis of frequency filtering decomposition preconditioners in this paper frequency filtering decomposition ffd preconditioner is analyzed by the approach of fourier analysis the condition number estimation of a preconditioned 2 d model problem is presented analysis reveals that condition number of the preconditioned matrix grows like o h 1 with h be the mesh size by using the framework of ffd a stabilized frequency filtering decomposition sffd method is proposed and analyzed by fourier method results show that sffd preconditioner is superior to ffd preconditioner in the sense that k m s f f d 1a k m f f d 1a numerical tests are performed to illustrate the theoretical results and the superiority of sffd preconditioner

modified tangential frequency filtering decomposition and its fourier analysis in this paper a modified tangential frequency filtering decomposition mtffd preconditioner is proposed the optimal order of the modification and the optimal relaxation parameter is determined by fourier analysis with the choice of optimal order of modification the fourier results show that the condition number of the preconditioned matrix is mathcal o h frac 2 3 and the spectrum distribution of the preconditioned matrix can be predicted by the fourier results the performance of mtffd preconditioner is compared with tangential frequency filtering tffd preconditioner on a variety of large sparse matrices arising from the discretization of pdes with discontinuous coefficients the numerical results show that the mtffd preconditioner is much more efficient than the tffd preconditioner

the generalized distance matrix of digraphs abstract let d g and d q g d i a g t r d g be the distance matrix and distance signless laplacian matrix of a simple strongly connected digraph g respectively where d i a g t r diag d 1 d 2 d n be the diagonal matrix with vertex transmissions of the digraph g to track the gradual change of d g into d q g in this paper we propose to study the convex combinations of d g and d i a g t r defined by d g d i a g t r 1 d g 0 1 this study reduces to merging the distance spectral and distance signless laplacian spectral theories the eigenvalue with the largest modulus of d g is called the d spectral radius of g denoted by g we determine the digraph which attains the maximum or minimum d spectral radius among all strongly connected digraphs moreover we also determine the digraphs which attain the minimum d spectral radius among all strongly connected digraphs with given parameters such as dichromatic number vertex connectivity or arc connectivity

on the multiplicity of α as an eigenvalue of the aα matrix of a graph in terms of the number of pendant vertices abstract let g v g e g be a simple undirected graph with vertex set v g and edge set e g the cyclomatic number of a connected graph g is defined as g e g v g 1 the a matrix of a graph g is defined by nikiforov as a g d g 1 a g where 0 1 a g and d g respectively denotes the adjacency matrix and the diagonal matrix of the vertex degrees of g let m g be the multiplicity of as an eigenvalue of a g a cluster of g is an independent set of one or more vertices of g each of which has the same set of neighbors the closure of a cluster is the set of vertices of the cluster together with all the neighbors of the cluster the degree of a cluster is the cardinality of its shared set of neighbors a k cluster is a cluster of degree k the number of vertices in a k cluster is its order a collection of two or more k clusters is independent if the k clusters are pairwise disjoint and is isolated if the closures of the k clusters are pairwise disjoint cardoso et al 2 obtained a lower bound for m g in terms of the number of pendant vertices they proved that for a simple connected graph g with p g 0 pendant vertices attached at q g quasi pendant vertices m g p g q g with equality if each internal vertex is a quasi pendant vertex in this article for a graph g of order n and with t independent k clusters of orders r 1 r t we prove that m g k i 1 t r i t and if the k clusters are isolated then m g k n 2 t which extends cardoso s result to a more general case also we give an upper bound for m g in terms of p g it is proved that m g p g 2 g with g the cyclomatic number of g the graphs g whose m g attain the upper bound are characterized explicitly

local fourier analysis for cell centered multigrid methods on triangular grids this work is focused on the design of efficient multigrid methods for discretizations on voronoi meshes associated with regular refinements of acute triangles for this purpose a special local fourier analysis for this type of discretizations is developed this analysis helps us to obtain appropriate novel smoothers depending on the geometry of the grid a two and three grid analysis are performed to analyze the behavior of the multigrid methods numerical test calculations validate the theoretical predictions furthermore the application of this analysis for the solution of more real problems by using semi structured grids is shown

geometric multigrid for the tight binding hamiltonian of graphene in order to calculate the electronic properties of graphene structures a tight binding approach can be used the tight binding formulation leads to linear systems of equations which are maximally i

fixed point theorems for nonlinear operators some new fixed point theorems are presented for operators of accretive nonlinear contractive or nonexpansive type these results are then used to establish a new existence principle for second order boundary value problems in hilbert spaces

the aα spread of a graph abstract let a g and d g be the adjacency matrix and the degree diagonal matrix of a graph g respectively for any real number 0 1 nikiforov defined the a matrix of a graph g as a g d g 1 a g the a spread of a graph is the difference between the largest eigenvalue and the smallest eigenvalue of the a matrix of the graph in this paper we obtain some new lower bounds on a spread and characterize the extremal graphs for certain cases which extend the results of a spread and q spread moreover some graph operations on a spread are presented

iterative and approximate solutions of general parameter dependent nonlinear operator equations in hilbert spaces iterative solutions of a general nonlinear operator equation of the form ax tx f where a and t are in general nonlinear operators in an appropriate space have not been developed so far in this paper the three well known banach mann and ishikawa iteration processes are used to find solutions of this equation in the hilbert space case an error estimate in each case is given the established results generalize many of the known results in literature

optimization of the multigrid convergence rate on semi structured meshes by local fourier analysis in this paper a local fourier analysis for multigrid methods on tetrahedral grids is presented different smoothers for the discretization of the laplace operator by linear finite elements on such grids are analyzed a four color smoother is presented as an efficient choice for regular tetrahedral grids whereas line and plane relaxations are needed for poorly shaped tetrahedra a novel partitioning of the fourier space is proposed to analyze the four color smoother numerical test calculations validate the theoretical predictions a multigrid method is constructed in a block wise form by using different smoothers and different numbers of pre and post smoothing steps in each tetrahedron of the coarsest grid of the domain some numerical experiments are presented to illustrate the efficiency of this multigrid algorithm

starlike trees are determined by their laplacian spectrum in this paper we show that if g is a starlike tree then it is determined by its laplacian spectrum moreover we prove some facts about trees with the same adjacency spectrum as a starlike tree

on the aα spectra of trees abstract let g be a graph with adjacency matrix a g and let d g be the diagonal matrix of the degrees of g for every real 0 1 define the matrix a g as a g d g 1 a g this paper gives several results about the a matrices of trees in particular it is shown that if t is a tree of maximal degree then the spectral radius of a t satisfies the tight inequality a t 2 1 1 which implies previous bounds of godsil lovasz and stevanovic the proof is deduced from some new results about the a matrices of bethe trees and generalized bethe trees in addition several bounds on the spectral radius of a of general graphs are proved implying tight bounds for paths and bethe trees

toward an h independent algebraic multigrid method for maxwell s equations we propose a new algebraic multigrid amg method for solving the eddy current approximations to maxwell s equations this amg method has its roots in an algorithm proposed by reitzinger and schoberl the main focus in the reitzinger and schoberl method is to maintain null space properties of the weak nabla times nabla times operator on coarse grids while these null space properties are critical they are not enough to guarantee h independent convergence rates of the overall multigrid scheme we present a new strategy for choosing intergrid transfers that not only maintains the important null space properties on coarse grids but also yields significantly improved multigrid convergence rates this improvement is related to those we explored in a previous paper but is fundamentally simpler easier to compute and performs better with respect to both multigrid operator complexity and convergence rates the new strategy builds on ideas in smoothed aggregation to improve the approximation property of an existing interpolation operator by carefully choosing the smoothing operators we show how it is sometimes possible to achieve h independent convergence rates with a modest increase in multigrid operator complexity though this ideal case is not always possible the overall algorithm performs significantly better than the original scheme in both iterations and run time finally the reitzinger and schoberl method as well as our previous smoothed method are shown to be special cases of this new algorithm

which graphs are determined by their spectrum for almost all graphs the answer to the question in the title is still unknown here we survey the cases for which the answer is known not only the adjacency matrix but also other types of matrices such as the laplacian matrix are considered

the characteristic polynomial of the laplacian of graphs in a b linear classes in this work we deal with the characteristic polynomial of the laplacian of a graph we present some general results about the coefficients of this polynomial we present families of graphs for which the number of edges m is given by a linear function of the number of vertices n in some of these graphs we can find certain coefficients of the above named polynomial as functions just of n

spectral characterizations of lollipop graphs abstract the lollipop graph denoted by h n p is obtained by appending a cycle c p to a pendant vertex of a path p n p we will show that no two non isomorphic lollipop graphs are cospectral with respect to the adjacency matrix it is proved that for p odd the lollipop graphs h n p and some related graphs h n p are determined by the adjacency spectrum and that all lollipop graphs are determined by its laplacian spectrum

an improved algebraic multigrid method for solving maxwell s equations we propose two improvements to the reitzinger and schoberl algebraic multigrid amg method for solving the eddy current approximations to maxwell s equations the main focus in the reitzinger schoberl method is to maintain null space properties of the weak nabla times nabla times operator on coarse grids while these null space properties are critical they are not enough to guarantee h independent convergence of the overall multigrid method we illustrate how the reitzinger schoberl amg method loses h independence due to the somewhat limited approximation property of the grid transfer operators we present two improvements to these operators that not only maintain the important null space properties on coarse grids but also yield significantly improved multigrid convergence rates the first improvement is based on smoothing the reitzinger schoberl grid transfer operators the second improvement is obtained by using higher order nodal interpolation to derive the corresponding amg interpolation operators while not completely h independent the resulting amg cg method demonstrates improved convergence behavior while maintaining low operator complexity

fourier analysis for multigrid methods on triangular grids in this paper a local fourier analysis technique for multigrid methods on triangular grids is presented the analysis is based on an expression of the fourier transform in new coordinate systems both in space variables and in frequency variables associated with reciprocal bases this tool makes it possible to study different components of the multigrid method in a very similar way to that of rectangular grids different smoothers for the discrete laplace operator obtained with linear finite elements are analyzed a new three color smoother has been studied and has proven to be the best choice for near equilateral triangles it is also shown that the block line smoothers are more appropriate for irregular triangles numerical test calculations validate the theoretical predictions

fixed point theory for generalized contractions on spaces with two metrics we present new fixed point results for generalized contractions on spaces with two metrics in addition generalized contractive homotopies will also be discussed in detail