a mesh connected area time optimal vlsi multiplier of large integers this paper describes a vlsi network for the multiplication of two n bit integers for very large n the network with its area 0 n and operation time 0 n matches within a constant factor the known theoretical n2 lower bound to the area time 2measure of complexity in the vlsi model of computation the network which is based on the discrete fourier transform has an extremely regular mesh structure and thus all wires have approximately the same length

integer division in linear time with bounded fan in a binary algorithm for division of an m n bit integer by an n bit integer is presented the algorithm produces the m 1 bit quotient and the n bit remainder in time o m n two hardware implementations one using combinational logic in cellular arrays and one employing systolic arrays are given these implementations are designed for modularity and regularity and thus are suitable for vlsi systems an important property of these implementations is that decisions are based on only one bit of the operands thus fan in and length of connecting wires are bounded independently of operand size in addition the systolic implementation has area o n which is the best possible

space time trade offs for higher radix modular multiplication using repeated addition the value of using a higher radix for modular multiplication in the context of rsa is investigated the main conclusion is that for algorithms which perform the multiplication via repeated addition there is broadly speaking a direct trade off between space and time provided by change of radix thus chip area utilized is roughly proportional to speed however initially as the radix is increased from 2 there is a short lived increase in speed greater than the extra area used

at sup 2 optimal galois field multiplier for vlsi vlsi designs for galois field multipliers which are central in many encoding and decoding procedures for error detecting and error correcting codes are presented an at sup 2 optimal galois field multiplier based on at sup 2 optimal integer multipliers for a synchronous vlsi model is exhibited galois field multiplication is done in two steps first two polynomials of degree n 1 over z sub p are multiplied and then the resulting polynomial is reduced modulo a fixed irreducible polynomial of degree n multiplication of polynomials is done by discrete fourier transform dft for p 2 the procedure is more involved for z sub p x than for z x an extension to the case of variable p is included and some open problems are stated

very simple methods for all pairs network flow analysis a very simple algorithm for the classical problem of computing the maximum network flow value between every pair of nodes in an undirected capacitated n node graph is presented as in the well known gomory hu method the method given here uses only n 1 maximum flow computations our algorithm is implemented by adding only five simple lines of code to any program that produces a minimum cut a program to produce an equivalent flow tree which is a compact representation of the flow values is obtained by adding only three simple lines of code to any program producing a minimum cut a very simple version of the gomory hu cut tree method that finds one minimum cut for every pair of nodes is also derived and it is shown that the seemingly fundamental operation of that method node contraction is not needed nor must crossing cuts be avoided as a result this version of the gomory hu method is implemented by adding less than ten simple lines of code to any program that produces a minimum cut the algor

on algebraic connectivity augmentation abstract suppose that g is an undirected graph and that h is a spanning subgraph of g c whose edges induce a subgraph on p vertices we consider the expression g h g where denotes the algebraic connectivity specifically we provide upper and lower bounds on g h g in terms of p and characterise the corresponding equality cases we also discuss the density of the expression g h g in the interval 0 p a bound on g h g is provided in a special case and several examples are considered

area time optimal vlsi integer multiplier with minimum computation time according to vlsi theory logn n is the range of computation times for which there may exist an at2 optimal multiplier of n bit integers such networks were previously known for the time range log2n 0 n in this paper we settle this theoretical question by exhibiting a class of at2 optimal multipliers with computation times logn 0 n1 2 our designs are based on the dft on a fermat ring whose elements are represented in a redundant radix 4 form to ensure 0 1 addition time

solution bases of multiterminal cut problems gomory and hu proved that the left n atop 2 right maxflow problems in an undirected network have at most n 1 distinct solution values we generalize their theorems in several ways an interesting special case is that the on4 2 commodity maxflow problems on an undirected network have at most left n atop 2 right distinct solution values these results follow from general theorems on the number of distinct solution values to sets of optimization problems we then show how the solutions can be compactly represented by a basis of the solution matrix a where aij 1 if solution j is feasible for problem i and aij 0 otherwise for the special case of the 2 cut problem we also show how to construct such a representation efficiently

the spectra of a graph obtained from copies of a generalized bethe tree abstract we generalize the concept of a bethe tree as follows we say that an unweighted rooted tree is a generalized bethe tree if in each level the vertices have equal degree if b k is a generalized bethe tree of k levels then we characterize completely the eigenvalues of the adjacency matrix and laplacian matrix of a graph b k r obtained from the union of r copies of b k and the cycle c r connecting the r vertex roots moreover we give results on the multiplicity of the eigenvalues on the spectral radii and on the algebraic conectivity

spectra of generalized bethe trees attached to a path abstract a generalized bethe tree is a rooted tree in which vertices at the same distance from the root have the same degree let p m be a path of m vertices let b i 1 i m be a set of generalized bethe trees let p m b i 1 i m be the tree obtained from p m and the trees b 1 b 2 b m by identifying the root vertex of b i with the i th vertex of p m we give a complete characterization of the eigenvalues of the laplacian and adjacency matrices of p m b i 1 i m in particular we characterize their spectral radii and the algebraic conectivity moreover we derive results concerning their multiplicities finally we apply the results to the case b 1 b 2 b m

normalized cuts and image segmentation we propose a novel approach for solving the perceptual grouping problem in vision rather than focusing on local features and their consistencies in the image data our approach aims at extracting the global impression of an image we treat image segmentation as a graph partitioning problem and propose a novel global criterion the normalized cut for segmenting the graph the normalized cut criterion measures both the total dissimilarity between the different groups as well as the total similarity within the groups we show that an efficient computational technique based on a generalized eigenvalue problem can be used to optimize this criterion we applied this approach to segmenting static images as well as motion sequences and found the results to be very encouraging

graphs with given diameter maximizing the algebraic connectivity we propose a class of graphs g d n1 n2 nd 1 containing of a chain of d 1 cliques kn1 kn2 knd 1 where neighboring cliques are fully interconnected the class of graphs has diameter d and size n 1 i d 1ni we prove that this class of graphs can achieve the maximal number of links the minimum average hopcount and more interestingly the maximal of any laplacian eigenvalue among all graphs with n nodes and diameter d the algebraic connectivity is the eigenvalue of the laplacian that has been studied most because it features many interesting properties we determine the graph with the largest algebraic connectivity among graphs with n nodes and diameter d 4 for other diameters numerically searching for the maximum of any eigenvalue is feasible because a the searching space within the class g d n1 n2 nd 1 is much smaller than within all graphs with n nodes and diameter d b we reduce the calculation of the laplacian spectrum from a n n to a d 1 d 1 matrix the maximum of any laplacian eigenvalue obtained either theoretically or by numerical searching is applied to 1 investigate the topological features of graphs that maximize different laplacian eigenvalues 2 study the correlation between the maximum algebraic connectivity amax n d and n as well as d and 3 evaluate two upper bounds of the algebraic connectivity that are proposed in the literature

efficient vlsi parallel algorithm for delaunay triangulation on orthogonal tree network in two and three dimensions an algorithm with worst case time complexity o log sup 2 n in two dimensions and o m sup 1 2 log n in three dimensions with n input points and m as the number of tetrahedra in triangulation is given its at sup 2 vlsi complexity on thompson s logarithmic delay model 1983 is o n sup 2 log sup 6 n in two dimensions and o m sup 2 nlog sup 4 n in three dimensions

the ordering of trees and connected graphs by algebraic connectivity abstract in this paper we first determine that the first four trees of order n 9 with the smallest algebraic connectivity are p n q n w n and z n with p n q n w n z n t where t is any tree of order n other than p n q n w n and z n then we consider the effect on the laplacian eigenvalues of connected graphs by suitably adding edges by using these results and methods we finally determine that the first six connected graphs of order n 9 with the smallest algebraic connectivity are p n q n q n w n w n and w n with p n q n q n w n w n w n g where g is any connected graph of order n other than p n q n q n w n w n and w n

o n depth modular exponentiation circuit algorithm an o n depth polynomial size combinational circuit algorithm is proposed for n bit modular exponentiation i e for the computation of x sup v mod m for arbitrary integers x y and m represented as n bit binary integers within bounds 2 sup n 1 spl les m 2 sup n and 0 spl les ix y m the algorithm is a generalization of the square and multiply method the terms x 2 sup l mod m s for all is spl epsiv 0 n 1 are computed in n 1 spl alpha logn parallel rounds each of which computes spl alpha logn consecutive terms where spl alpha spl ges 1 logn the circuit implementing a round has depth o 1 spl alpha logn and size o n sup 2 1 spl alpha yielding a circuit for modular exponentiation of depth o 1 spl alpha spl alpha n and size o n sup 3 2 spl alpha spl alpha logn

the effect on the algebraic connectivity of a tree by grafting or collapsing of edges abstract let g v e be a tree on n 2 vertices and let v v let l g be the laplacian matrix of g and g be its algebraic connectivity let g k l be the graph obtained from g by attaching two new paths p vv 1 v 2 v k and q vu 1 u 2 u l of length k and l respectively at v we prove that if l k 1 then g k 1 l 1 g k l let v 1 v 2 be an edge of g let g be the tree obtained from g by deleting the edge v 1 v 2 and identifying the vertices v 1 and v 2 then we prove that g g as a corollary to the above results we obtain the celebrated theorem on algebraic connectivity which states that among all trees on n vertices the path has the smallest and the star has the largest algebraic connectivity

systolic multipliers for finite fields gf 2 m two systolic architectures are developed for performing the product sum computation ab c in the finite field gf 2m of 2melements where a b and c are arbitrary elements of gf 2m the first multiplier is a serial in serial out one dimensional systolic array while the second multiplier is a parallel in parallel out two dimensional systolic array the first multiplier requires a smaller number of basic cells than the second multiplier the second multiplier heeds less average time per computation than the first multiplier if a number of computations are performed consecutively to perform single computations both multipliers require the same computational time in both cases the architectures are simple and regular and possess the properties of concurrency and modularity as a consequence they are well suited for use in vlsi systems

a vlsi modulo m multiplier a novel method to compute the exact digits of the modulo m product of integers is proposed and a modulo m multiply structure is defined such a structure can be implemented by means of a few fast vlsi binary multipliers and a response time of about 150 200 ns to perform modular multiplications with moduli up to 32767 can be reached a comparison to rom based structures is also provided the modular multiplier has been evaluated asymptotically according to the vlsi complexity theory and it turned out to be an optimal design this structure can be used to implement a residue multiplier in arithmetic structures using residue number systems rnss the complexity of this residue multiplier has been evaluated and lower complexity figures than for rom based multiply structures have been obtained under several hypotheses on rns parameters

some results on the laplacian eigenvalues of unicyclic graphs abstract in this paper we provide the smallest value of the second largest laplacian eigenvalue for any unicyclic graph and find the unicyclic graphs attaining that value and also give an asymptotically good upper bounds for the second largest laplacian eigenvalues of unicyclic graphs using this results we can determine unicyclic graphs with maximum laplacian separator and unicyclic graphs with maximum laplacian spread will also be determined

algebraic connectivity and degree sequences of trees we investigate the structure of trees that have minimal algebraic connectivity among all trees with a given degree sequence we show that such trees are caterpillars and that the vertex degrees are non decreasing on every path on non pendant vertices starting at the characteristic set of the fiedler vector author s abstract

on the sum of the laplacian eigenvalues of a tree given an n vertex graph g v e the laplacian spectrum of g is the set of eigenvalues of the laplacian matrix l d a where d and a denote the diagonal matrix of vertex degrees and the adjacency matrix of g respectively in this paper we study the laplacian spectrum of trees more precisely we find a new upper bound on the sum of the k largest laplacian eigenvalues of every n vertex tree where k 1 n this result is used to establish that the n vertex star has the highest laplacian energy over all n vertex trees which answers affirmatively to a question raised by radenkovic and gutman 10

λt complexity measures for vlsi computations in constant chip area the computational complexity measures introduced here are motivated by the trend to higher vlsi integration levels rather than increased chip area to accomplish solutions to larger problem instances it seems that the increase in the computational power of vlsi circuits can be mainly attributed to the reduction in the minimum feature size rather than to an increase in the chip area in view of this we present a constant area perspective and consider the discrete fourier transform and related problems in a vlsi model that has and t as its resources advantages of the mesh algorithm over the shuffle exchange algorithm in the computation time for the dft are shown to arise from an upper bound on current density in the wires which we suggest must be considered in any vlsi grid model

graphs of given order and size and minimum algebraic connectivity the structure of connected graphs of given size and order that have minimal algebraic connectivity is investigated it is shown that they must consist of a chain of cliques moreover an upper bound for the number of maximal cliques of size 2 or larger is derived

new efficient structure for a modular multiplier for rns modular multiplication is a very important arithmetic operation in residue based real time computing systems in realizing these multipliers rom based structures are more efficient for small moduli due to the exponential growth of rom sizes implementations with arithmetic components are more suitable for medium and large moduli this paper presents a new modular multiplier that can deal efficiently with medium and large size moduli the design of this modular multiplier that multiplies two n bit residue digits consists basically of a n spl times n binary multiplier a n 1 k spl times k binary multiplier k n three n bit adders and a small size combinational circuit when compared with the most competitive published work the new multiplier reduces significantly both time delay and hardware requirements the design is very suitable for vlsi realization

on the ordering of trees by the laplacian coefficients abstract we generalize the results from x d zhang x p lv y h chen ordering trees by the laplacian coefficients linear algebra appl 2009 doi 10 1016 j laa 2009 04 018 on the partial ordering of trees with given diameter for two n vertex trees t 1 and t 2 if c k t 1 c k t 2 holds for all laplacian coefficients c k k 0 1 n we say that t 1 is dominated by t 2 and write t 1 c t 2 we proved that among n vertex trees with fixed diameter d the caterpillar c n d has minimal laplacian coefficients c k k 0 1 n the number of incomparable pairs of trees on 18 vertices is presented as well as infinite families of examples for two other partial orderings of trees recently proposed by mohar for every integer n we construct a chain t i i 0 m of n vertex trees of length n 2 4 such that t 0 s n t m p n and t i c t i 1 for all i 0 1 m 1 in addition the characterization of the partial ordering of starlike trees is established by the majorization inequalities of the pendent path lengths we determine the relations among the extremal trees with fixed maximum degree and with perfect matching and further support the laplacian coefficients as a measure of branching

the influence of miroslav fiedler on spectral graph theory abstract this note is a write up of a talk given at the ilas meeting in braunschweig 2011 at the minisymposium celebrating the 80th birthday of miroslav fiedler the purpose of the talk is to outline the impact of fiedler s work on the development of spectral graph theory fiedler is best known for putting forward the algebraic connectivity and its eigenvector these two topics were genuine gold strikes which have motivated thousands of studies in pure and applied science but there is more in fiedler s work that influences spectral graph theory and still more waiting to be discovered

the algebraic connectivity of lollipop graphs abstract let c n g be the lollipop graph obtained by appending a g cycle c g to a pendant vertex of a path on n g vertices in 2002 fallat kirkland and pati proved that for n 3 g 1 2 and g 4 c n g c n g 1 in this paper we prove that for g 4 c n g c n g 1 for all n where c n g is the algebraic connectivity of c n g

bipartite modular multiplication method this paper proposes a new modular multiplication method that uses montgomery residues defined by a modulus m and a montgomery radix r whose value is less than the modulus m this condition enables the operand multiplier to be split into two parts that can be processed separately in parallel increasing the calculation speed the upper part of the split multiplier can be processed by calculating a product modulo m of the multiplicand and this part of the split multiplier the lower part of the split multiplier can be processed by calculating a product modulo m of the multiplicand this part of the split multiplier and the inverse of a constant r two different implementations based on this method are proposed one uses a classical modular multiplier and a montgomery multiplier and the other generates partial products for each part of the split multiplier separately which are added and accumulated in a single pipelined unit a radix 4 version of a multiplier based on a radix 4 classical modular multiplier and a radix 4 montgomery multiplier has been designed and simulated the proposed method is also suitable for software implementation in a multiprocessor environment

ordering of the trees with a perfect matching by minimal energies the ordering of the trees with a perfect matching according to minimal energies is investigated using a new method proposed here we obtain the preceding trees in the increasing order of their energies within the class under consideration for n 3 where 2n is the vertex number of the tree we deduce the first 11 12 13 14 12 9 16 and 21 trees for n 21 15 n 20 n 14 10 n 13 n 9 n 8 n 7 and n 6 respectively for n 5 4 3 we list all the trees in the increasing order of their energies

on graphs with equal algebraic and vertex connectivity let g be an undirected unweighted graph on n vertices let l be its laplacian matrix and let l l i j be the group inverse of l it is known that for z l 1 2 max1 i j n s 1n li s lj s the quantity 1 z l is a lower bound on the algebraic connectivity a g of g while the vertex connectivity of g v g is an upper bound on a g we characterize the graphs g for which v g a g and subsequently prove that if n v g 2 then v g a g holds if and only if 1 z l a g v g we close with an example showing that the equality 1 z l a g does not necessarily imply that 1 z l a g v g

practical cellular dividers a discussion is presented of parallel division algorithms that can be classified among modified higher radix nonrestoring online division methods where redundant representations are extensively utilized to speed up the operation the network realizations of these algorithms are cellular or even systolic with exclusively local control they have both size area and time of o n where n is the length of the dividend representation the same structures can also be used as a signed digit serial multiplier when suitably equipped with some control and a few registers the divider multiplier brings remarkable performance to large modular arithmetic rsa cryptography and greatest common divisor computations they are also of interest for the design of floating point units and signal processing applications

ordering of huckel trees according to minimal energies abstract the tree with a perfect matching having degrees not greater than three is referred to as the huckel tree the ordering of huckel trees according to their minimal energies is investigated by means of a quasi ordering relation using a simpler method than that of li h li on minimal energy ordering of acyclic conjugated molecules j math chem 25 1999 145 169 we obtain the first 2 n 10 1 1 n 2 trees in the increasing order of their energies within the class under consideration for n 13 where 2 n is the vertex number of the tree the number of the trees obtained here exceeds the reported result li 1999 by n 10 we also get a lot of preceding trees in the increasing order of their energies within the class for 5 n 12

ordering trees with algebraic connectivity and diameter abstract this paper is motivated by abreu s paper which proposed finding a total ordering of trees by their algebraic connectivity is still an open problem we first discuss how to find a total ordering of trees with diameter 4 then these results are used to determine first several orderings of trees by algebraic connectivity in nonincreasing order in last we present a class of trees with large diameter whose algebraic connectivity is larger than that of a tree with diameter 3

dynamic programming and graph optimization problems abstract several classes of graph optimization problems which can be solved using dynamic programming are known to have more efficient tailor made algorithms this paper discusses four such classes and the underlying constraints on their subproblem interrelationships that yield these efficient algorithms these classes are also extended to handle more general cost functions

the algebraic connectivity of graphs under perturbation in this paper we investigate how the algebraic connectivity of a connected graph behaves when the graph is perturbed by separating or grafting an edge

triangulations tilings and certain block triangular matrices for a convex polygonp withn sides a partitioning ofp inton 2 nonoverlapping triangles each of whose vertices is a vertex ofp is called a triangulation or tiling and each triangle is a tile each tile has a given cost associated with it which may differ one from another this paper considers the problem of finding a tiling ofp such that the sum of the costs of the tiles used is a minimum and explores the curiosity that an abstract formulation of it can be cast as a linear program further the special structure of the linear program permits a recursive o n 3 algorithm

logarithmic depth circuits for algebraic functions this paper describes circuits for computation of a large class of algebraic functions on polynomials power series and integers for which it has been a long standing open problem to compute in depth less than omega log n 2 algebraic circuits assume unit cost for elemental addition and multiplication this paper describes o log n depth algebraic circuits which given as input the coefficients of n degree polynomials over an appropriate ring compute the product of n o 1 polynomials the symmetric functions as well as division and interpolation of real polynomials also described are o log n depth algebraic circuits which are given as input the first n coefficients of a power series over an appropriate ring compute the product of n o 1 power series as well as division reciprocal and reversion of real power series furthermore this paper describes boolean circuits of depth o log n log log n which given n bit binary numbers compute the product of n numbers and integ

spectra of weighted generalized bethe trees joined at the root abstract a generalized bethe tree is a rooted tree in which vertices at the same distance from the root have the same degree let b i 1 i m be a set of trees such that for i 1 2 m 1 b i is a generalized bethe tree of k i levels 2 the vertices of b i at the level j have degree d i k i j 1 for j 1 2 k i and 3 the edges of b i joining the vertices at the level j with the vertices at the level j 1 have weight w i k i j for j 1 2 k i 1 let v b i 1 i m be the tree obtained from the union of the trees b i joined at their respective root vertices we give a complete characterization of the eigenvalues of the laplacian and adjacency matrices of v b i 1 i m moreover we derive results concerning their multiplicities in particular we characterize the spectral radii the algebraic conectivity and the second largest laplacian eigenvalue

ordering trees by their laplacian spectral radii denote by tn the set of trees on n vertices zhang li x d zang j s li the two largest eigenvalues of laplacian matrices of trees in chinese j china univ sci technol 28 1998 513 518 and guo j m guo on the laplacian spectral radius of a tree linear algebra appl 368 2003 379 385 give the first four trees in tn ordered according to their laplacian spectral radii in this paper we determine the fifth to eighth trees in the above ordering

an efficient parallel dynamic programming algorithm abstract antonio tsai and huang proposed a scheme in 1991 to parallelize the standard dynamic programming approach to solve combinatorial multistage problems however their dynamic programming approach is restricted to those multistage problems where the decision made at each stage depends only on decisions made in the stage immediately preceding it for many interesting problems the decision at each stage depends on the decisions made at all the previous stages and therefore their approach doesn t apply the matrix chain multiplication problem longest common subsequence problem and optimal polygon triangulation problem are some examples of such problems we also present techniques for parallelizing the dynamic programming solution to such problems the parallel algorithm we develop for a pram has complexity n employing n 2 processors since the traditional sequential algorithm for such problems is n 3 our parallel algorithm is an optimal parallel algorithm based on this traditional algorithm we also describe the results of our experiments that are in conformity with our theoretical complexity results we also compare and contrast our result with results obtained by earlier researchers and show that our parallel algorithm has optimal efficiency of 100 with respect to the traditional dynamic programming algorithm

the limit points of laplacian spectra of graphs abstract let g be a graph on n vertices denote by l g the laplacian matrix of g it is easy to see that l g is positive semidefinite symmetric and that its second smallest eigenvalue g 0 if and only if g is connected this observation let fiedler to call g the algebraic connectivity of the graph g in this paper the limit points of laplacian spectra of graphs are investigated particular attention is given to the limit points of algebraic connectivity some new results and generalizations are included

a carry free algorithm for finding the greatest common divisor of two integers abstract we investigate a variant of the so called binary algorithm for finding the gcd greatest common divisor of two numbers which requires no comparisons we show that when implemented with carry save hardware it can be used to find the modulo b inverse of an n bit binary integer in a time proportional to n using only registers of length proportional to n such a hardware implementation of this algorithm set up for finding inverses with respect to a 336 bit modulus b would have applications in the currently expanding field of secure data transmission and storage in such an implementation multiplication in linear time both modulo b and ordinary would come along as a by product because multiplication can be achieved by a sequence of nine inversions some additions and negations

vlsi architectures for computing multiplications and inverses in gf 2 m finite field arithmetic logic is central in the implementation of reed solomon coders and in some cryptographic algorithms there is a need for good multiplication and inversion algorithms that can be easily realized on vlsi chips massey and omura 1 recently developed a new multiplication algorithm for galois fields based on a normal basis representation in this paper a pipeline structure is developed to realize the massey omura multiplier in the finite field gf 2m with the simple squaring property of the normal basis representation used together with this multiplier a pipeline architecture is also developed for computing inverse elements in gf 2m the designs developed for the massey omura multiplier and the computation of inverse elements are regular simple expandable and therefore naturally suitable for vlsi implementation

combinatorial perron parameters for trees abstract the notion of combinatorial perron value was introduced in 2 we continue the study of this parameter and also introduce a new parameter e m which gives a new lower bound on the spectral radius of the bottleneck matrix m of a rooted tree we prove a bound on the approximation error for e m several properties of these two parameters are shown these ideas are motivated by the concept of algebraic connectivity a certain extension property for the combinatorial perron value is shown and it allows us to define a new center concept for caterpillars we also compare computationally this new center to the so called characteristic set i e the center obtained from algebraic connectivity

on eigenfunctions of markov processes on trees we begin by studying the eigenvectors associated to irreducible finite birth and death processes showing that the i nontrivial eigenvector i admits a succession of i decreasing or increasing stages each of them crossing zero imbedding naturally the finite state space into a continuous segment one can unequivocally define the zeros of i which are interlaced with those of i 1 these kind of results are deduced from a general investigation of minimax multi sets dirichlet eigenproblems which leads to a direct construction of the eigenvectors associated to birth and death processes this approach can be generically extended to eigenvectors of markov processes living on trees this enables to reinterpret the eigenvalues and the eigenvectors in terms of the previous dirichlet eigenproblems and a more general conjecture is presented about related higher order cheeger inequalities finally we carefully study the geometric structure of the eigenspace associated to the spectral gap on trees

log depth circuits for division and related problems we present optimal depth boolean circuits depth o log n for integer division powering and multiple products we also show that these three problems are of equivalent uniform depth and space complexity in addition we describe an algorithm for testing divisibility that is optimal for both depth and space

ordering trees and graphs with few cycles by algebraic connectivity abstract several approaches for ordering graphs by spectral parameters are presented in the literature we can find graph orderings either by the greatest eigenvalue spectral radius or index or by the sum of the absolute values of the eigenvalues the energy of a graph or by the second smallest eigenvalue of the laplacian matrix the algebraic connectivity among others by considering the fact that the algebraic connectivity is related to the connectivity and shape of the graphs several structural properties of graphs relative to this parameter have been studied hence a large number of papers about ordering graphs by algebraic connectivity mainly about trees and graphs with few cycles have been published this paper surveys the significant results concerning these topics trying to focus on possible points to be investigated in order to understand the difficulties to obtain partial orderings via algebraic connectivity

ordering trees by their largest eigenvalues abstract the set of trees with n vertices is denoted by t n hofmeister has determined the first five values of the largest eigenvalue of trees in t n and the corresponding trees for these values linear algebra appl 260 1997 43 in other words an order of the first five trees in t n by their largest eigenvalues has been given focus on the same purpose we shall give a partition for trees in t n first in the this paper and then extend this order to the eighth tree

ordering trees by their largest eigenvalues abstract let t and 1 t denote the maximum degree and the largest eigenvalue of a tree t respectively let t n be the set of trees on n vertices and t n t t n t in the present paper among the trees in t n n 4 we characterize the tree which alone minimizes the largest eigenvalue as well as the tree which alone maximizes the largest eigenvalue when n 2 2 n 1 furthermore it is proved that for two trees t1 and t2 in t n n 4 if t 1 2 n 3 1 and t1 t2 then 1 t1 1 t2 by applying this result we extend the order of trees in t n by their largest eigenvalues to the 13th tree when n 12 this extends the results of hofmeister linear algebra appl 260 1997 43 and chang et al linear algebra appl 370 2003 175

area time optimal division for t ω log n 1 e abstract area time optimal vlsi division circuits are described for all computation times in the range log n 1 e o n for arbitrary e 0

a vlsi residue arithmetic multiplier recently residue arithmetic has received increased attention in the open literature using table lookup methods and high speed memory modular arithmetic has been demonstrated however the memory size limitation of ecl bipolar and high speed mos limits the admissible size of the moduli used in the numbering system in this paper the moduli size limitation is overcome using vlsi technology special architectures and moduli choice a residue multiplier having a 48 72 bit dynamic range capable of performing 10m multiplication s is reported

spectral partitioning works planar graphs and finite element meshes abstract spectral partitioning methods use the fiedler vector the eigenvector of the second smallest eigenvalue of the laplacian matrix to find a small separator of a graph these methods are important components of many scientific numerical algorithms and have been demonstrated by experiment to work extremely well in this paper we show that spectral partitioning methods work well on bounded degree planar graphs and finite element meshes the classes of graphs to which they are usually applied while naive spectral bisection does not necessarily work we prove that spectral partitioning techniques can be used to produce separators whose ratio of vertices removed to edges cut is o n for bounded degree planar graphs and two dimensional meshes and o n1 d for well shaped d dimensional meshes the heart of our analysis is an upper bound on the second smallest eigenvalues of the laplacian matrices of these graphs we prove a bound of o 1 n for bounded degree planar graphs and o 1 n2 d for well shaped d dimensional meshes

ordering trees by algebraic connectivity abstract the trees of order n 15 and algebraic connectivity no less than 2 3 have been recently classified in this paper we determine all trees of order n 45 with algebraic connectivity in the interval 5 21 2 2 3

ordering of trees with fixed matching number by the laplacian coefficients abstract let g be a simple undirected graph with the characteristic polynomial of its laplacian matrix l g p g x k 0 n 1 k c k x n k aleksandar ilic a ilic trees with minimal laplacian coefficients comput math appl 59 2010 2776 2783 identified n vertex trees with given matching number q which simultaneously minimize all laplacian coefficients in this paper we give another proof of this result generalizing the approach in the above paper we determine n vertex trees with given matching number q which have the second minimal laplacian coefficients we also identify the n vertex trees with a perfect matching having the largest and the second largest laplacian coefficients respectively extremal values on some indices such as wiener index modified hyper wiener index laplacian like energy incidence energy of n vertex trees with matching number q are obtained in this paper

on the two largest eigenvalues of trees abstract very little is known about upper bounds for the largest eigenvalues of a tree that depend only on the vertex number starting from a classical upper bound for the largest eigenvalue some refinements can be obtained by successively removing trees from consideration the results can be used to characterize those trees that maximize the second largest eigenvalue this corrects a result from the literature and it includes a proof of a conjecture of neumaier the main tool for this endeavor is the theory of partial engenvectors

the arithmetic cube we present the design of a vlsi processor which can be programmed to compute the discrete fourier transform of a sequence of n points and which achieves the theoretical at2 lower bound of n2 for n n where n is an infinite set furthermore since the set n is also sufficiently dense the processor achieves for any n the theoretical at2 lower bound of n2 for computing the cyclic convolution of two sequences of n points uniquely our design achieves this bound without the use of data shuffling or long wires also the processor uses only approximately n multipliers while many other designs need n multipliers to achieve the same time bounds since multipliers are usually much larger than adders the processor presented in this paper should be smaller the design also features layout regularity minimal control and nearest neighbor interconnect of arithmetic cells of a few different types these characteristics make it an ideal candidate for vlsi implementation

a combinatorial limit to the computing power of vlsi circuits we introduce a property of boolean functions called transitivity which consists of integer polynomial and matrix products as well as of many interesting related computational problems we show that the area of any circuit computing a transitive function grows quadratically with the circuit s maximum data rate expressed in bits s this result provides a precise analytic expression of an area time tradeoff for a wide class of vlsi circuits furthermore as shown elsewhere this tradeoff is achievable we have thus matching to within a constant multiplicative factor upper and lower complexity bounds for the three above products in the vlsi circuits computational model

on the algebraic connectivity of some caterpillars a sharp upper bound and a total ordering a caterpillar is a tree in which the removal of all pendant vertices makes it a path let d 3 and n 2 d 1 be given let p p1 p2 pd 1 with p1 1 p2 1 pd 1 1 let c p be the caterpillar obtained from the stars sp1 sp2 spd 1 and the path pd 1 by identifying the root of spi with the i vertex of pd 1 let c c p p1 p2 pd 1 n d 1 we prove that the algebraic connectivity of c p c is bounded above by 124 2 4 8 2cos d 2 d 1 moreover we prove that if d is even then c p p 1 1 p d2 1 1 p d2 n 2d 3 is the unique caterpillar in c attaining the upper bound and that if d is odd then the upper bound cannot be achieved finally for 1 k d 12 we give a total ordering by algebraic connectivity on ck c1 1 pk 1 1 pd k 1 1 c pk pd k

a note on limit points for algebraic connectivity abstract we consider the limit points for algebraic connectivity i e those real numbers r such that r is the limit of a sequence of distinct numbers each of which is the algebraic connectivity of some graph we prove that each nonnegative real number is a limit point for algebraic connectivity and characterize the limit points for algebraic connectivity of trees

characterizing trees with large laplacian energy we investigate the problem of ordering trees according to their laplacian energy more precisely given a positive integer n we find a class of cardinality approximately p n whose elements are the n vertex trees with largest laplacian energy the main tool for establishing this result is a new upper bound on the sum sk t of the k largest laplacian eigenvalues of an n vertex tree t with diameter at least four where k 2 f1 ng

exponentiation using division chains exponentiation may be performed faster than the traditional square and multiply method by iteratively reducing the exponent module numbers which as exponents themselves require few multiplications this mainly includes those with few nonzero bits for a suitable choice of such divisors the resulting mixed basis representation of the exponent reduces the expected number of nonsquaring multiplications by over half at the cost of a single extra register preprocessing effort depends entirely on the exponent and can be kept down to the work saved in a single exponentiation moreover no precomputed look up tables are required so the method is especially applicable where space is at a premium in particular it outperforms the instance of the m ary method which uses the same space however for 512 bit exponents it beats every instance of the m ary method achieving well under 635 multiplications on average both hardware and software implementations of the rsa crypto system can benefit from this algorithm

modular multiplication without trial division let n 1 we present a method for multiplying two integers called n residues modulo n while avoiding division by n n residues are represented in a nonstandard way so this method is useful only if several computations are done modulo one n the addition and subtraction algorithms are unchanged 1 description some algorithms 1 2 4 5 require extensive modular arith metic we propose a representation of residue classes so as to speed modular multiplication without affecting the modular addition and subtraction algorithms other recent algorithms for modular arithmetic appear in 3 6 fix n 1 define an a residue to be a residue class modulo n select a radix r coprime to n possibly the machine word size or a power thereof such that r n and such that computations modulo r are inexpensive to process let r l and n be integers satisfying 0 n then return t n else return t to validate redc observe mn tn n tmod r so t is an integer also tr tmod n so t tr x mod n thirdly 0 t mn rn rn so 0 t 2n if r and n are large then t mn may exceed the largest double precision value one can circumvent this by adjusting m so r m 0 given two numbers x and y between 0 and n 1 inclusive let z redc xy then z xy r x mod n so xr l yr x zrx mod n also 0 z n so z is the product of x and y in this representation other algorithms for operating on n residues in this representation can be derived from the algorithms normally used the addition algorithm is unchanged since xr x yr x zr x mod n if and only if x y z mod n also unchanged are

computation of matrix chain products part ii this paper considers the computation of matrix chain products of the form m 1 times m 2 times cdots times m n 1 if the matrices are of different dimensions the order in which the product is computed affects the number of operations an optimum order is an order which minimizes the total number of operations we present some theorems about an optimum order of computing the matrices based on these theorems an o n log n algorithm for finding an optimum order will be presented in part ii

an autoscale residue multiplier dynamic range overflow is a serious problem in residue arithmetic systems contemporary overflow management schemes rely on inefficient scaling operations in this correspondence a residue scaler is architectured which inhibits dynamic range overflow the system uses the popular three moduli set 2n 1 2n 2n 1 using a 4k memory model practical 12 and 18 bit autoscalers are configured an error model for the derived residue arithmetic unit is also derived and experimentally verified