on the largest kth eigenvalues of trees abstract we consider the only remaining unsolved case n 0 mod k for the largest k th eigenvalue of trees with n vertices we give complete solutions for the cases k 2 3 4 5 and give some necessary conditions for extremal trees in general cases

w tsv weighted topological signature vector for lexicon reduction in handwritten arabic documents this paper proposes a holistic lexicon reduction method for ancient and modern handwritten arabic documents the word shape is represented by the weighted topological signature vector w tsv which encodes graph data into a low dimensional vector space three directed acyclic graph dag representations are proposed for arabic word shapes based on topological and geometrical features lexicon reduction is achieved by a nearest neighbors search in the w tsv space the proposed framework has been tested on the ifn enit and the ibn sina databases achieving respectively a degree of reduction of 83 5 and 92 9 for an accuracy of reduction of 90

a deletion contraction algorithm for the characteristic polynomial of a multigraph expression du polynome caracteristique d un multigraphe fini g en termes de polynomes caracteristiques des modifications locales de g application de la formule obtenue a l etude des plus grandes valeurs propres de certains graphes theta

a note on the multiplicities of graph eigenvalues abstract let g be a graph with vertex set 1 n and let h be the graph obtained by attaching one pendant path of length k i at vertex i i 1 r 1 r n for a real symmetric matrix a whose graph is h let m a denote the multiplicity of an eigenvalue of a from a result in da fonseca 2005 7 we know that m a n in this note we characterize the case m a n we also give two upper bounds on eigenvalue multiplicity of trees and unicyclic graphs which are generalizations of some results in rowlinson 2010 10

graphs for small multiprocessor interconnection networks abstract let d be the diameter of a graph g and let 1 be the largest eigenvalue of its 0 1 adjacency matrix we give a proof of the fact that there are exactly 69 non trivial connected graphs with d 1 1 9 these 69 graphs all have up to 10 vertices and were recently found to be suitable models for small multiprocessor interconnection networks we also examine the suitability of integral graphs to model multiprocessor interconnection networks especially with respect to the load balancing problem in addition we classify integral graphs with small values of d 1 1 in connection with the load balancing problem for multiprocessor systems

rigorous sensitivity analysis for parameter dependent systems of equations abstract in this paper we construct rigorous and realistic enclosures for the solutions of parameter dependent systems of equations f x t 0 the enclosure of the solution x x t is given in the form x 0 s t t 0 with interval vectors x0 t0 and an interval matrix s this allows a detailed analysis of the sensitivity of the solution x t to small changes of t and the influence of specific changes in selected parameters on the solution the linear case is treated separately

the kth largest eigenvalue of a tree abstract let k t be the k th eigenvalue of a tree x the largest integer not greater than x it is shown that a tree with n vertices has k t n 2 k for 2 k n 2 and this upper bound is best possible for n 1 mod k

on graphs whose star sets are co cliques abstract in this paper we study graphs all of whose star sets induce cliques or co cliques we show that the star sets of every tree for each eigenvalue are independent sets among other results it is shown that each star set of a connected graph g with three distinct eigenvalues induces a clique if and only if g k 1 2 or k 2 2 it is also proved that stars are the only graphs with three distinct eigenvalues having a star partition with independent star sets

relational duals of the c means clustering algorithms abstract the hard and fuzzy c means algorithms are widely used effective tools for the problem of clustering n objects into hard or fuzzy groups of similar individuals when the data is available as object data consisting of a set of n feature vectors in rp however object data algorithms are not directly applicable when the n objects are implicitly described in terms of relational data which consists of a set of n2 measurements of relations between each of the pairs of objects new relational versions of the hard and fuzzy c means algorithms are presented here for the case when the relational data can reasonably be viewed as some measure of distance some convergence properties of the algorithms are given along with a numerical example

rigorous sensitivity analysis for systems of linear and nonlinear equations methods are presented for performing a rigorous sensitivity analysis of numerical problems with independent noncorrelated data for general systems of linear and nonlinear equations the methods may serve for the following two purposes first to bound the dependency of the solution on changes in the input data in contrast to condition numbers a componentwise sensitivity analysis of the solution vector is performed second to estimate the true solution set for problems the input data of which are afflicted with tolerances the methods presented are very effective with the additional property that due to an automatic error control mechanism every computed result is guaranteed to be correct examples are given for linear systems demonstrating that the computed bounds are in general very sharp interesting comparisons to traditional condition numbers are given

k means clustering based data detection and symbol timing recovery for burst mode optical receiver burst mode receivers are key components of optical transmission systems including passive optical networks and have received much attention in recent years we present new efficient methods for burst optical signal detection in burst mode data transmission using a modified k means clustering technique we also develop a data aided feedforward symbol timing recovery method based on a polynomial interpolation and maximum likelihood estimation theory a performance criterion considering the error caused by the interpolation approximation is derived for this method the proposed detection and timing recovery approaches can be implemented effectively and rapidly therefore they are very suitable for burst mode receivers we also provide some numerical examples to demonstrate the performance of the proposed methods

the spectral radius of trees on k pendant vertices abstract in this paper we consider the following problem of all trees of order n with k pendant vertices n k fixed which achieves the maximal spectral radius we show that the maximal spectral radius is obtained uniquely at t n k where t n k is a tree obtained from a star k 1 k and k paths of almost equal lengths by joining each pendant vertex of k 1 k to an end vertex of one path we also discuss the spectral radius of t n k and get some results

the graphs with spectral radius between 2 and 2 5 abstract we complete the determination of the graphs in the title begun by cvetkovic doob and gutman

graphs with least eigenvalue 2 ten years on abstract the authors monograph spectral generalizations of line graphs was published in 2004 following the successful use of star complements to complete the classification of graphs with least eigenvalue 2 guided by citations of the book we survey progress in this area over the past decade some new observations are included

new neutrosophic approach to image segmentation neutrosophic set ns a part of neutrosophy theory studies the origin nature and scope of neutralities as well as their interactions with different ideational spectra ns is a formal framework that has been recently proposed however ns needs to be specified from a technical point of view for a given application or field we apply ns after defining some concepts and operations for image segmentation the image is transformed into the ns domain which is described using three membership sets t i and f the entropy in ns is defined and employed to evaluate the indeterminacy two operations a mean and b enhancement operations are proposed to reduce the set indeterminacy finally the proposed method is employed to perform image segmentation using a c means clustering we have conducted experiments on a variety of images the experimental results demonstrate that the proposed approach can segment the images automatically and effectively especially it can segment the clean images and the images having noise with different noise levels

a study of eigenspaces of graphs abstract we investigate the relationship between the structure of a graph and its eigenspaces the angles between the eigenspaces and the vectors of the standard basis of r n play an important role the key notion is that of a special basis for an eigenspace called a star basis star bases enable us to define a canonical bases of r n associated with a graph and to formulate an algorithm for graph isomorphism

maximal reflexive cacti with four cycles the approach via smith graphs abstract cacti or treelike graphs are graphs whose all cycles are mutually edge disjoint graphs with the property 2 2 are called reflexive graphs where 2 is the second largest eigenvalue of the corresponding 0 1 adjacency matrix the property 2 2 is a hereditary one i e all induced subgraphs of a reflexive graph are also reflexive this is why we represent reflexive graphs through the maximal graphs within a given class such as connected cacti with a fixed number of cycles in previous work we have determined all maximal reflexive cacti with four cycles whose cycles do not form a bundle and pointed out the role of so called pouring of smith graphs in their construction in this paper besides pouring we show several other patterns of the appearance of smith trees in those constructions these include splitting of a smith tree adding an edge to a smith tree and then splitting of the resulting graph identifying two vertices of a smith tree and then splitting the resulting graph our results show that the presence of smith trees is evident in all such maximal reflexive cacti with four cycles and that in most of them smith graphs appear in the described way

adaptive spatial information theoretic clustering for image segmentation the incorporation of spatial context into clustering algorithms for image segmentation has recently received a significant amount of attention many modified clustering algorithms have been proposed and proven to be effective for image segmentation in this paper we propose a different framework for incorporating spatial information with the aim of achieving robust and accurate segmentation in case of mixed noise without using experimentally set parameters based on the original robust information clustering ric algorithm called adaptive spatial information theoretic clustering asic algorithm the proposed objective function has a new dissimilarity measure and the weighting factor for neighborhood effect is fully adaptive to the image content it enhances the smoothness towards piecewise homogeneous segmentation and reduces the edge blurring effect furthermore a unique characteristic of the new information segmentation algorithm is that it has the capabilities to eliminate outliers at different stages of the asic algorithm these result in improved segmentation result by identifying and relabeling the outliers in a relatively stronger noisy environment comprehensive experiments and a new information theoretic proof are carried out to illustrate that our new algorithm can consistently improve the segmentation result while effectively handles the edge blurring effect the experimental results with both synthetic and real images demonstrate that the proposed method is effective and robust to mixed noise and the algorithm outperforms other popular spatial clustering variants

on the spectral radius of graphs with cut edges we study the spectral radius of graphs with n vertices and k cut edges in this paper we show that of all the connected graphs with n vertices and k cut edges the maximal spectral radius is obtained uniquely at k k where k k n is a graph obtained by joining k independent vertices to one vertex of kn k we also discuss the limit point of the maximal spectral radius

constructing trees with given eigenvalues and angles abstract let eigenvalues and angles between eigensubspaces and the axes of the corresponding real vector space be given for a tree an algorithm for constructing all trees with given eigenvalues and angles is described

a characterization of oriented hypergraphic laplacian and adjacency matrix coefficients abstract an oriented hypergraph is an oriented incidence structure that generalizes and unifies graph and hypergraph theoretic results by examining its locally signed graphic substructure in this paper we obtain a combinatorial characterization of the coefficients of the characteristic polynomials of oriented hypergraphic laplacian and adjacency matrices via a signed hypergraphic generalization of basic figures of graphs additionally we provide bounds on the determinant and permanent of the laplacian matrix characterize the oriented hypergraphs in which the upper bound is sharp and demonstrate that the lower bound is never achieved

the second largest eigenvalue of a tree denote by 2 t the second largest eigenvalue of a tree t an easy algorithm is given to decide whether 2 t for a given number and a structure theorem for trees with 2 t is proved also it is shown that a tree t with n vertices has 2 t lsqb n 3 2rsqb12 this bound is best possible for odd n

convergence properties of the generalized fuzzy c means clustering algorithms fuzzy c means fcm clustering algorithms have been widely used to solve clustering problems yang and yu 1 extended these to optimization procedures with respect to any probability distribution they showed that the optimal cluster centers are the fixed points of these generalized fcm clustering algorithms the convergence properties of algorithms are the important theoretical issue in this paper we present convergence properties of the generalized fcm clustering algorithms these are global convergence local convergence and its rate of convergence

ncm neutrosophic c means clustering algorithm abstract in this paper a new clustering algorithm neutrosophic c means ncm is introduced for uncertain data clustering which is inspired from fuzzy c means and the neutrosophic set framework to derive such a structure a novel suitable objective function is defined and minimized and the clustering problem is formulated as a constrained minimization problem whose solution depends on the objective function in the objective function two new types of rejection have been introduced the ambiguity rejection which concerns the patterns lying near the cluster boundaries and the distance rejection dealing with patterns that are far away from all the clusters these measures are able to manage uncertainty due to imprecise and or incomplete definition of the clusters we conducted several experiments with synthetic and real data sets the results are encouraging and compared favorably with results from other methods as fcm pcm and fpcm algorithms on the same data sets finally the proposed method was applied into image segmentation algorithm the experimental results show that the proposed algorithm can be considered as a promising tool for data clustering and image processing

general type 2 fuzzy c means algorithm for uncertain fuzzy clustering pattern recognition in real world data is subject to various sources of uncertainty that should be appropriately managed the focus of this paper is the management of uncertainty associated with parameters of fuzzy clustering algorithms type 2 fuzzy sets t2 fss have received increased research interest over the past decade primarily due to their potential to model various uncertainties however because of the computational intensity of the processing of general t2 fuzzy sets gt2 fss only their constrained version i e the interval t2 it2 fss were typically used fortunately the recently introduced concepts of planes and zslices allow for efficient representation and computation with gt2 fss following this recent development this paper presents a novel approach for uncertain fuzzy clustering using the general type 2 fuzzy c means gt2 fcm algorithm the proposed method builds on top of the previously published it2 fcm algorithm which is extended via the planes representation theorem the fuzzifier parameter of the fcm algorithm can be expressed using linguistic terms such as small or high which are modeled as t1 fss this linguistic fuzzifier value is then used to construct the gt2 fcm cluster membership functions the linguistic uncertainty is transformed into uncertain fuzzy positions of the extracted clusters the gt2 fcm algorithm was found to balance the performance of t1 fcm algorithms in various uncertain pattern recognition tasks and to provide increased robustness in situations where noisy or insufficient training data are present

a 1 0 1 and sparsest basis for the null space of a forest in optimal time abstract given a matrix the null space problem asks for a basis of its null space having the fewest nonzeros this problem is known to be np complete and even hard to approximate the null space of a forest is the null space of its adjacency matrix sander and sander 2005 and akbari et al 2006 independently proved that the null space of each forest admits a 1 0 1 basis we devise an algorithm for determining a sparsest basis of the null space of any given forest which in addition is a 1 0 1 basis our algorithm is time optimal in the sense that it takes time at most proportional to the number of nonzeros in any sparsest basis of the null space of the input forest moreover we show that given a forest f on n vertices the set of those vertices x for which there is a vector in the null space of f that is nonzero at x and the number of nonzeros in any sparsest basis of the null space of f can be found in o n time

self validating methods abstract we present some ideas on self validating sv methods this note is especially intended to give some background for the articles in this special volume of laa it is not intended to be a survey of the big variety of all possible aspects of sv methods but rather a summary of some basic concepts especially some common misconceptions are mentioned and explored

local discretization error bounds using interval boundary element method in this paper a method to account for the point wise discretization error in the solution for boundary element method is developed interval methods are used to enclose the boundary integral equation and a sharp parametric solver for the interval linear system of equations is presented the developed method does not assume any special properties besides the laplace equation being a linear elliptic partial differential equation whose green s function for an isotropic media is known numerical results are presented showing the guarantee of the bounds on the solution as well as the convergence of the discretization error copyright 2008 john wiley sons ltd

fuzzy sets as a basis for a theory of possibility the theory of possibility described in this paper is related to the theory of fuzzy sets by defining the concept of a possibility distribution as a fuzzy restriction which acts as an elastic constraint on the values that may be assigned to a variable more specifically if f is a fuzzy subset of a universe of discourse u u which is characterized by its membership function f then a proposition of the form x is f where x is a variable taking values in u induces a possibility distribution tx which equates the possibility of x taking the value u to f u the compatibility of u with f in this way x becomes a fuzzy variable which is associated with the possibility distribution tx in much the same way as a random variable is associated with a probability distribution in general a variable may be associated both with a possibility distribution and a probability distribution with the weak connection between the two expressed as the possibility probability consistency principle a thesis advanced in this paper is that the imprecision that is intrinsic in natural languages is in the main possibilistic rather than probabilistic in nature thus by employing the concept of a possibility distribution a proposition p in a natural language may be translated into a procedure which computes the probability distribution of a set of attributes which are implied by p several types of conditional translation rules are discussed and in particular a translation rule for propositions of the form x is f is possible where is a number in the interval 0 1 is formulated and illustrated by examples

ecmdd evidential c medoids clustering with multiple prototypes in this work a new prototype based clustering method named evidential c medoids ecmdd which belongs to the family of medoid based clustering for proximity data is proposed as an extension of fuzzy c medoids fcmdd on the theoretical framework of belief functions in the application of fcmdd and original ecmdd a single medoid prototype which is supposed to belong to the object set is utilized to represent one class for the sake of clarity this kind of ecmdd using a single medoid is denoted by secmdd in real clustering applications using only one pattern to capture or interpret a class may not adequately model different types of group structure and hence limits the clustering performance in order to address this problem a variation of ecmdd using multiple weighted medoids denoted by wecmdd is presented unlike secmdd in wecmdd objects in each cluster carry various weights describing their degree of representativeness for that class this mechanism enables each class to be represented by more than one object experimental results in synthetic and real data sets clearly demonstrate the superiority of secmdd and wecmdd moreover the clustering results by wecmdd can provide richer information for the inner structure of the detected classes with the help of prototype weights

line star sets for laplacian eigenvalues abstract let g be a mixed graph with a nonzero laplacian eigenvalue of multiplicity k a line star set for in g is a set y of k edges of g such that is not a laplacian eigenvalue of g y it is shown that line star set exists for any nonzero laplacian eigenvalue of any mixed graph some basic properties for line star sets are given we also obtain some results on line star sets in undirected graphs

notes on the second largest eigenvalue of a graph abstract for a fixed real number r we give several necessary and or sufficient conditions for a graph to have the second largest eigenvalue of the adjacency matrix or signless laplacian matrix less then or equal to r

constructing fullerene graphs from their eigenvalues and angles we discuss means of constructing fullerene graphs from their eigenvalues and angles an algorithm for such a construction is given

a conjecture on the second largest eigenvalue of a tree with perfect matchings abstract in this paper we present an upper bound for the second largest eigenvalue of a tree on n 2 k 4 t t 2 vertices with perfect matchings at the same time the few largest second largest eigenvalues over the class of trees on n 2 k 4 t t 2 vertices with perfect matchings are also obtained

star partitions and regularity in graphs abstract we investigate regular graphs with a star partition x 1 x 2 x m such that g xi is regular for some i the cubic graphs with this property are explored in detail

the fuzzy c 2 means solving the ambiguity rejection in clustering abstract in this paper we deal with the clustering problem whose goal consists of computing a partition of a family of patterns into disjoint classes the method that we propose is formulated as a constrained minimization problem whose solution depends on a fuzzy objective function in which reject options are introduced two types of rejection have been included the ambiguity rejection which concerns patterns lying near the class boundaries and the distance rejection dealing with patterns that are far away from all the classes to compute these rejections we propose an extension of the fuzzy c means fcm algorithm of bezdek pattern recognition with fuzzy objective function algorithms plenum press new york 1981 this algorithm is called the fuzzy c 2 means fc 2m these measures allow to manage uncertainty due both to imprecise and incomplete definition of the classes the advantages of our method are 1 the degree of membership to the reject classes for a pattern xk are learned in the iterative clustering problem 2 it is not necessary to compute other characteristics to determine the reject and ambiguity degrees 3 the partial ambiguity rejections introduce a discounting process between the classical fcm membership functions in order to avoid the memberships to be spread across the classes 4 the membership functions are more immune to noise and correspond more closely to the notion of compatibility preliminary computational experiences on the developed algorithm are encouraging and compared favorably with results from other methods as fcm fpcm and f c 1 m fuzzy c 1 means clustering with solely distance rejection algorithms on the same data sets the differences in the performance can be attributed to the fact that ambiguous patterns are less accounted in for the computing of the centers

overestimation in linear interval equations bounds are given for the overestimation of the solution set of a system of linear interval equations by various inclusion intervals for this set in particular a theorem of gay this journal 19 1982 pp 858 870 on the quadratic approximation property of the preconditioned fixpoint inverse is strengthened

indexing hierarchical structures using graph spectra hierarchical image structures are abundant in computer vision and have been used to encode part structure scale spaces and a variety of multiresolution features in this paper we describe a framework for indexing such representations that embeds the topological structure of a directed acyclic graph dag into a low dimensional vector space based on a novel spectral characterization of a dag this topological signature allows us to efficiently retrieve a promising set of candidates from a database of models using a simple nearest neighbor search we establish the insensitivity of the signature to minor perturbation of graph structure due to noise occlusion or node split merge to accommodate large scale occlusion the dag rooted at each nonleaf node of the query votes for model objects that share that part effectively accumulating local evidence in a model dag s topological subspaces we demonstrate the approach with a series of indexing experiments in the domain of view based 3d object recognition using shock graphs

on the spectral radius of quasi tree graphs abstract a connected graph g v e is called a quasi tree if there exists u 0 v g such that g u 0 is a tree denote q n d 0 g g is a quasi tree graph of order n with g u 0 being a tree and d g u 0 d 0 in this paper we determined the maximal and the second maximal spectral radii of all quasi tree graphs in the set q n d 0

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

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

nerf c means non euclidean relational fuzzy clustering abstract the relational fuzzy c means rfcm algorithm can be used to cluster a set of n objects described by pair wise dissimilarity values if and only if there exist n points in r n 1 whose squared euclidean distances precisely match the given dissimilarity data this strong restriction on the dissimilarity data renders rfcm inapplicable to most relational clustering problems this paper substantially improves rfcm by generalizing it to the case of arbitrary symmetric dissimilarity data the generalization is obtained using a computationally efficient modification of the existing algorithm that is equivalent to applying a spreading transformation to the dissimilarity data while the method given applies specifically to dissimilarity data a simple transformation can be used to convert similarity relations into dissimilarity data so the method is applicable to any numerical relational data that are positive reflexive or anti reflexive and symmetric numerical examples illustrate and compare the present approach to problems that can be studied with alternatives such as the linkage algorithms

ecm an evidential version of the fuzzy c means algorithm a new clustering method for object data called ecm evidential c means is introduced in the theoretical framework of belief functions it is based on the concept of credal partition extending those of hard fuzzy and possibilistic ones to derive such a structure a suitable objective function is minimized using an fcm like algorithm a validity index allowing the determination of the proper number of clusters is also proposed experiments with synthetic and real data sets show that the proposed algorithm can be considered as a promising tool in the field of exploratory statistics

a posteriori error estimates for boundary element methods this paper deals with a general framework for a posteriori error estimates in boundary element methods which is specified for three examples namely symm s integral equation an integral equation with a hypersingular operator and a boundary integral equation for a transmission problem based on these estimates an analog of eriksson and johnson s adaptive finite element method is proposed for the h version of the galerkin boundary element method for integral equations of the first kind the efficiency of the approach is shown by numerical experiments which yield almost optimal convergence rates even in the presence of singularities the construction of an adaptive mesh refinement procedure is of very high practical importance in the numerical analysis of partial differential equations and we refer to the pioneering work of babuska and miller 3 and eriksson and johnson 10 11 whereas the main features ofadaptivity for finite element methods now seem to be visible and the door is open to implementation 15 comparably little is known for boundary element methods for integral equations see e g 1 13 18 19 24 in this paper a new adaptive version of the galerkin discretization for the boundary element method is presented based on a posteriori error estimates a general framework for these a posteriori error estimates is derived in 2 and three examples are discussed in 3 5 involving the dirichlet problem the neumann problem for a closed and an open surface and a transmission problem for the laplacian leading to integral equations with strongly elliptic pseudodifferential operators even for smooth data the lack of regularity of the solution near corners of a polygonal domain q leads to poor solutions of the numerical schemes unless appropriate singular functions are incorporated in the trial space or a suitable mesh refinement is used in practical problems such information is missing e g when we have singular or nearly singular data and the main problem is how to balance a graded mesh refinement towards singularities and a global

axiomatic generalization of the membership degree weighting function for fuzzy c means clustering for decades practitioners have been using the center based partitional clustering algorithms like fuzzy c means fcm which rely on minimizing an objective function comprising of an appropriately weighted sum of distances of each data point from the cluster representatives numerous generalizations in terms of choice of the distance function have been introduced for fcm since its inception in a stark contrast to this fact to the best of our knowledge there has not been any significant effort to address the issue of convergence of the algorithm under the structured generalization of the weighting function here by generalization we mean replacing the conventional weighting function uijm where uij indicates the membership of data xi to cluster cj and m is the real valued fuzzifier with 1 m with a more general function g uij which can assume a wide variety of forms under some mild assumptions in this article for the first time we present a novel axiomatic development of the general weighting function based on the membership degrees we formulate the fuzzy clustering problem along with the intuitive justification of the technicalities behind the axiomatic approach we develop an alternative optimization ao procedure to solve the main clustering problem by solving the partial optimization problems in an iterative way the novelty of the article lies in the development of an axiomatic generalization of the weighting function of fcm formulation of an ao algorithm for the fuzzy clustering problem with the extension to this general class of weighting functions and a detailed convergence analysis of the proposed algorithm

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

combinatorial approach for computing the characteristic polynomial of a matrix the study of the spectrum of weighted di graphs has attracted recently much attention in the literature here we use the coates digraph as a main tool to extend in a combinatorial way some well known results from the spectral graph theory on computing the characteristic polynomial of graphs new results are related to weighted di graph and thus to any square matrix

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

sharp upper bounds on the second largest eigenvalues of connected graphs abstract let 2 be the second largest eigenvalue of a graph powers 1988 4 gave some upper bounds of 2 for general graphs and bipartite graphs respectively considering that these bounds are not always attainable for connected graphs we present sharp upper bounds of 2 for connected graphs and connected bipartite graphs in this paper moreover the extremal graphs are completely characterized

on graphs whose signless laplacian index does not exceed 4 5 abstract let a g and d g be respectively the adjacency matrix and the degree matrix of a graph g the signless laplacian matrix of g is defined as q g d g a g the q spectrum of g is the set of the eigenvalues together with their multiplicities of q g the q index of g is the maximum eigenvalue of q g the possibilities for developing a spectral theory of graphs based on the signless laplacian matrices were discussed by cvetkovic et al d cvetkovic p rowlinson s k simic signless laplacians of finite graphs linear algebra appl 423 2007 155 171 in the latter paper the authors determine the graphs whose q index is in the interval 0 4 in this paper we investigate some properties of q spectra of graphs especially for the limit points of the q index by using these results we characterize respectively the structures of graphs whose the q index lies in the intervals 4 2 5 2 5 2 and 2 4 5 where 1 3 54 6 33 1 3 54 6 33 1 3 2 382975767

worst case bounds on the point wise discretization error in boundary element method for the elasticity problem in this work point wise discretization error is bounded via interval approach for the elasticity problem using interval boundary element formulation the formulation allows for computation of the worst case bounds on the boundary values for the elasticity problem from these bounds the worst case bounds on the true solution at any point in the domain of the system can be computed examples are presented to demonstrate the effectiveness of the treatment of local discretization error in elasticity problem via interval methods

a possibilistic fuzzy c means clustering algorithm in 1997 we proposed the fuzzy possibilistic c means fpcm model and algorithm that generated both membership and typicality values when clustering unlabeled data fpcm constrains the typicality values so that the sum over all data points of typicalities to a cluster is one the row sum constraint produces unrealistic typicality values for large data sets in this paper we propose a new model called possibilistic fuzzy c means pfcm model pfcm produces memberships and possibilities simultaneously along with the usual point prototypes or cluster centers for each cluster pfcm is a hybridization of possibilistic c means pcm and fuzzy c means fcm that often avoids various problems of pcm fcm and fpcm pfcm solves the noise sensitivity defect of fcm overcomes the coincident clusters problem of pcm and eliminates the row sum constraints of fpcm we derive the first order necessary conditions for extrema of the pfcm objective function and use them as the basis for a standard alternating optimization approach to finding local minima of the pfcm objective functional several numerical examples are given that compare fcm and pcm to pfcm our examples show that pfcm compares favorably to both of the previous models since pfcm prototypes are less sensitive to outliers and can avoid coincident clusters pfcm is a strong candidate for fuzzy rule based system identification

on the spectral radius of trees abstract in this paper we investigated the spectral radius of trees and obtained the following result let t be a tree on n vertices let m t denote one maximum matching of t and m t i let t i be a tree as shown in fig 1 then t 1 2 n i 1 n i 1 2 4 n 2i 1 and equality holds if and only if t t i

bounds on graph eigenvalues abstract upper and lower estimates are found for the maximum of the kth eigenvalue of a graph as a function of the number of vertices or edges