spectra of digraphs in this primarily expository paper we survey classical and some more recent results on the spectra of digraphs equivalently the spectra of 0 1 matrices with emphasis on the spectral radius

a theoretical and empirical study on unbiased boundary extended crossover for real valued representation we present a new crossover operator for real coded genetic algorithms employing a novel methodology to remove the inherent bias of pre existing crossover operators this is done by transforming the topology of the hyper rectangular real space by gluing opposite boundaries and designing a boundary extension method for making the fitness function smooth at the glued boundary we show the advantages of the proposed crossover by comparing its performance with those of existing ones on test functions that are commonly used in the literature and a nonlinear regression on a real world dataset

skew adjacency matrices of graphs the spectra of the skew adjacency matrices of a graph are considered as a possible way to distinguish adjacency cospectral graphs this leads to the following topics graphs whose skew adjacency matrices are all cospectral relations between the matchings polynomial of a graph and the characteristic polynomials of its adjacency and skew adjacency matrices skew spectral radii and an analogue of the perron frobenius theorem and the number of skew adjacency matrices of a graph with distinct spectra

an adaptive parameter tuning of particle swarm optimization algorithm an adaptive parameter tuning of particle swarm optimization based on velocity information apso vi algorithm is proposed in this paper the velocity convergence of particles is first analyzed and the relationship between the velocity of particle and the search failures is pointed out which reveals the reasons why pso has relative poor global searching ability then this algorithm introduces the velocity information which is defined as the average absolute value of velocity of all the particles a new strategy is presented that the inertia weight is dynamically adjusted according to average absolute value of velocity which follows a given nonlinear ideal velocity by feedback control which can avoid the velocity closed to zero at the early stage under the guide of the nonlinear ideal velocity apso vi can maintain appropriate swarm diversity and alleviate the premature convergence validly numerical experiments are conducted to compare the proposed algorithm with different variants of pso on some benchmark functions experimental results show that the proposed algorithm remarkably improves the ability of pso to jump out of the local optima and significantly enhance the convergence speed and precision

a piecewise linear chaotic map and sequential quadratic programming based robust hybrid particle swarm optimization this paper presents a novel robust hybrid particle swarm optimization rhpso based on piecewise linear chaotic map pwlcm and sequential quadratic programming sqp the aim of the present research is to develop a new single objective optimization approach which requires no adjustment of its parameters for both unconstrained and constrained optimization problems this novel algorithm makes the best of ergodicity of pwlcm to help pso with the global search while employing the sqp to accelerate the local search five unconstrained benchmarks eighteen constrained benchmarks and three engineering optimization problems from the literature are solved by using the proposed hybrid approach the simulation results compared with other state of art methods demonstrate the effectiveness and robustness of the proposed rhpso for both unconstrained and constrained problems of different dimensions

an upper bound for the energy of radial digraphs abstract let d be a digraph with n vertices a arcs c 2 closed walks of length 2 and spectral radius d recently ayyaswamy balachandran and gutman 2011 1 proved that when d a c 2 2 n 1 it is possible to construct an upper bound for the energy of digraphs which improves the mcclelland inequality for the energy of strongly connected digraphs given in rada 2009 16 it is our interest in this paper to show that for general digraphs the inequality d a c 2 2 n does not hold however we introduce the class of radial digraphs that satisfy the spectral radius condition above and for this class of digraphs we improve the bound for the energy given in ayyaswamy et al 2011 1

mpso median oriented particle swarm optimization particle swarm optimization pso is a bio inspired optimization algorithm which has been empirically demonstrated to perform well on many optimization problems however it has two main weaknesses which have restricted the wider applications of pso the algorithm can easily get trapped in the local optima and has slow convergence speed therefore improvement and or elimination of these disadvantages are the most important objective in pso research in this paper we propose median oriented particle swarm optimization mpso to carry out a global search over entire search space with accelerating convergence speed and avoiding local optima the median position of particles and the worst and median fitness values of the swarm are incorporated in the standard pso to achieve the mentioned goals the proposed algorithm is evaluated on 20 unimodal multimodal rotated and shifted high dimensional benchmark functions and the results are compared with some well known pso algorithms in the literature the results show that mpso substantially enhances the performance of the pso paradigm in terms of convergence speed and finds global or good near global optimal in the functions

on almost regular tournament matrices abstract spectral and determinantal properties of a special class m n of 2n 2n almost regular tournament matrices are studied in particular the maximum perron value of the matrices in this class is determined and shown to be achieved by the brualdi li matrix which has been conjectured to have the largest perron value among all tournament matrices of even order we also establish some determinantal inequalities for matrices in m n and describe the structure of their associated walk spaces

on tournament matrices abstract let t be a tournament of order n with adjacency matrix m we find several conditions that are equivalent to m being singular a correlation between the number of 3 cycles in t and the rank of m is established it is shown that asymptotically at least 1 2 of the tournament matrices are nonsingular we also derive bounds on the spectral radius of tournament matrices with a given row sum vector

self adaptive mix of particle swarm methodologies for constrained optimization abstract in recent years many different variants of the particle swarm optimizer pso for solving optimization problems have been proposed however pso has an inherent drawback in handling constrained problems mainly because of its complexity and dependency on parameters furthermore one pso variant may perform well for some test problems but not obtain good results for others in this paper our purpose is to develop a new pso algorithm that can efficiently solve a variety of constrained optimization problems it considers a mix of different pso variants each of which evolves with a different number of individuals from the current population in each generation the algorithm assigns more individuals to the better performing variants and fewer to the worse performing ones also a new pso variant is developed for use in the proposed algorithm to maintain a better balance between its local and global pso versions a new methodology for adapting pso parameters is presented and the proposed self adaptive pso algorithm tested and analyzed on two sets of test problems namely the cec2006 and cec2010 constrained optimization problems based on the results the proposed algorithm shows significantly better performance than the same global and local pso variants as well as other state of the art algorithms although based on our analysis it cannot guarantee an optimal solution for any unknown problem it is expected to be able to solve a wide variety of practical problems

spectral radii of tournament matrices whose graphs are related by an arc reversal abstract given an irreducible tournament matrix t and a pair of distinct indices i and j let t i j be the matrix obtained from t by transposing its principal submatrix on rows and columns i and j we establish one condition on rows i and j of t under which the spectral radius of t i j is no smaller than that of t and another condition on the i th and j th entries of the left and right perron vectors of t under which the spectral radius of t i j must be strictly smaller than that of t these conditions are used to compare the spectral radii of a class of toeplitz tournament matrices and the resulting comparison sheds light on some conjectures of brualdi and li further if t yields equality in a certain lower bound on the spectral radius of a tournament matrix then for any i and j we provide simple necessary and sufficient conditions for the spectral radius of t i j to be larger than that of t to be smaller than that of t and to be equal to that of t

properties of the brualdi li tournament matrix the brualdi li tournament matrix is conjectured to have the largest spectral radius among all tournament matrices of even order in this paper two forms of the characteristic polynomial of the brualdi li tournament matrix are found using the first form it is shown that the roots of the characteristic polynomial are simple and that the brualdi li tournament matrix is diagonalizable using the second form an expression is found for the coefficients of the powers of the variable in the characteristic polynomial these coefficients give information about the cycle structure of the cycles of length 1 5 of the directed graph associated with the brualdi li tournament matrix

perron vector ordering for a subclass of tournament matrices abstract we consider the class of n n tournament matrices having scor vector 1 1 2 3 n 4 n 3 n 2 n 2 1 for each such matrix we give formulas for the entries in its perron vector in terms of the corresponding perron value this leads to a discussion of the ordering of the entries in the perron vector and so yields some insight into the kendall wei method for ranking players in a round robin competition finally we characterize all matrices from this class such that the ordering of the entries in the perron vector coincides with the ordering of the entries in the score vector

the bandwidth minimization problem for cyclic caterpillars with hair length 1 is np complete in this paper we show that the bandwidth minimization problem remains np complete for cyclic caterpillars with hair length 1 cyclic caterpillars with hair length 1 are graphs in which the removal of all pendant vertices results in a simple cycle

preserving regular tournaments and term rank 1 we investigate linear operators which map certain types of tournaments to themselves to this end we also characterize term rank 1 preservers on the set of matrices whose associated digraphs are simple loopless directed graphs and find that this set of operators is more diverse than might be expected

a decentralized quantum inspired particle swarm optimization algorithm with cellular structured population this paper proposes a decentralized form of quantum inspired particle swarm optimization qpso with cellular structured population called cqpso for keeping the population diversity and balancing the global and local search the cqpso is further improved by re designing the local attractor in the sub population called cqpso lbest in order to accelerate the diffusion of the best solution and thus enhance the performance of cqpso the particles in cqpso and cqpso lbest are distributed in a two dimensional 2d grid and only allowed to interact with their neighbors according to the specified neighborhood which plays a role in exploiting the search space inside the neighborhood the overlapping particles work for delivering the information among the nearest neighborhoods acting as exploring the search space with diffusion of solutions during the evolutionary process theoretical studies are made to analyze the global convergence of cpso and cqpso lbest based on the theory of probabilistic metric space we systematically investigate the performance of cqpso lbest on 42 benchmark functions with different properties including unimodal multimodal separated shifted rotated noisy and mis scaled and compare with a set of pso variants with different topologies and swarm based evolutionary algorithms eas the experimental results demonstrate the better performance of cqpso lbest moreover two real world problems which are two dimensional 2d iir digital filter design and economic dispatch ed problem from power systems area are used to evaluate cqpso lbest and the experimental results verified the advantages of cqpso lbest

some results on h hypertournament matrices abstract in this work we find a necessary and sufficient condition for the normality of an h hypertournament matrix moreover we give a sufficient condition for n 1 2 to be the spectral radius of a normal h hypertournament matrix of order n also we answer an open question suggested by kirkland

group inverses of m matrices associated with nonnegative matrices having few eigenvalues abstract we continue here earlier investigations of the structure of group generalized inverses a of singular and irreducible m matrices a ri m where m is an n n nonnegative and irreducible matrix whose spectral radius is r and where m is assumed to belong to one of several special classes of nonnegative matrices we now focus on cases where m has only a few distinct eigenvalues we are particularly interested in instances of such matrices m which cause all or almost all of the off diagonal entries of the corresponding a to be nonpositive we then apply our results to the study of the sign pattern of the off diagonal entries of the group generalized inverse of a ri m where m comes from one of the following families of nonnegative matrices 1 the magic square of order n 4k generated by matlab 2 the doubly regular tournament matrices the latter type is an example of a special type of a nonnegative matrix m for which the group inverse of the associated m matrix satisfies a a t for some 0

finding the minimum bandwidth of an interval graph abstract an assignment of unique integers to the vertices of a graph is called a linear layout the bandwidth minimization problem bandwidth is the following given a graph g v e and an integer k determine whether there exists a linear layout of g such that the maximum difference between adjacent vertices is bounded by k interval graphs are the intersection graphs of a family of intervals of the real line bandwidth remains np complete even when restricted to special subclasses of trees we show that bandwidth can be solved in time o n2 for interval graphs moreover for a given interval graph a linear layout with minimum bandwidth can be constructed in time o n2logn as a by product we get that this construction can be done for proper interval graphs in time o nlog n m

on the location of the spectrum of hypertournament matrices inclusion regions for the spectrum of a hypertournament matrix a are obtained based on a complex curve that relates the real and imaginary parts of the eigenvalues these results generalize and in certain cases improve the work of s kirkland linear and multilinear algebra 30 1991 261 the bounds obtained depend on the variance of the score vector their tightness is investigated using the notion of numerical range 2001 elsevier science inc all rights reserved

algebraic multiplicity of the eigenvalues of a tournament matrix let f denote the set of irreducible n x n tournament matrices here arc our main results 1 for all n 3 every matrix in k has at least three distinct eigenvalues such a matrix has exactly three distinct eigenvalues if and only if it is a hadamard tournament matrix 2 for all n 2 3 there is a matrix in y having n distinct eigenvalues 3 if cr denotes the maximum algebraic multiplicity of 0 as an eigenvalue of the matrices in z then ln 21 2 6 4 if r is the minimum perron value i e spectral radius of all matrices in r then 2 8

the skew energy of tournaments abstract let s be the skew adjacency matrix of a digraph the skew energy of s denoted by e s is the sum of the absolute values of the all eigenvalues of s in this paper for skew adjacency matrices of tournaments of order n we will give lower and upper bounds of the skew energy we show that our lower bound is tight in the sense that it coincides with the minimum of e s of all skew adjacency matrices of tournaments of order n in addition we will introduce the skew energy which is a generalization of the skew energy

adaptive fuzzy particle swarm optimization for global optimization of multimodal functions this paper proposes an adaptive fuzzy pso afpso algorithm based on the standard particle swarm optimization spso algorithm the proposed afpso utilizes fuzzy set theory to adjust pso acceleration coefficients adaptively and is thereby able to improve the accuracy and efficiency of searches incorporating this algorithm with quadratic interpolation and crossover operator further enhances the global searching capability to form a new variant called afpso qi we compared the proposed afpso and its variant afpso qi with spso quadratic interpolation pso qipso unified pso upso fully informed particle swarm fips dynamic multi swarm pso dmspso and comprehensive learning pso clpso across sixteen benchmark functions the proposed algorithms performed well when applied to minimization problems for most of the multimodal functions considered

extremal digraphs with given clique number abstract we classify those digraphs with a given number of vertices and a given clique number that maximize the perron root of the adjacency matrix

normal matrices and their principal submatrices of co order one abstract let a be a normal matrix v be any of its indices a v be the matrix obtained from a by deleting the v th row and column and be an eigenvalue of a v in our paper we construct the eigenspace of a associated with from that of a v in particular it is shown that if there is a unique jordan block of size strictly greater than one in the part of the jordan form of a v corresponding to then the geometric multiplicity of decreases by one under the transition from a v to a in other words the typical change of the spectral properties holds for the results obtained are applied to circulant matrices moreover in appendix to our paper we consider almost regular tournament matrices as principal submatrices of co order one of regular tournament matrices in particular it is observed that the brualdi li tournament matrix b 2 n of order 2n is permutationally similar to a principal submatrix of co order one of the circulant matrix of order 2n 1 with the first row 0 1 1 n 0 0 n as a consequence of this fact the weak brualdi li conjecture is formulated for principal submatrices of co order one of the adjacency matrices of cayley tournaments

on finding the minimum bandwidth of interval graphs abstract let g v e be an interval graph with v n and e m an o n 2 log n algorithm was proposed in kratsch inform comput 74 140 158 1987 to find the bandwidth of g we show that this algorithm is wrong and provide a corrected version of the same also it was observed in 4 that the bandwidth of a proper interval graph can be computed in o n log n m time we show how this idea can be modified slightly to yield an o n m algorithm

on the minimum perron value for an irreducible tournament matrix abstract perron values of tournament matrices have been of interest to a number of authors recently in 1983 brualdi and li made a conjecture as to which matrix minimizes the perron value over the class of irreducible n n tournament matrices we prove that conjecture using both matrix theoretic and graph theoretic techniques

memetic binary particle swarm optimization for discrete optimization problems in recent decades many researchers have been interested in algorithms inspired by the observation of natural phenomena to solve optimization problems among them meta heuristic algorithms have been extensively applied in continuous real and discrete binary search spaces such algorithms are appropriate for global searches because of their global exploration and local exploitation abilities in this study a memetic binary particle swarm optimization bpso scheme is introduced based on hybrid local and global searches in bpso the algorithm binary hybrid topology particle swarm optimization bhtpso is used to solve the optimization problems in the binary search spaces in addition a variant of the proposed algorithm binary hybrid topology particle swarm optimization quadratic interpolation bhtpso qi is proposed to enhance the global searching capability these algorithms are tested on two set of problems in the binary search space several nonlinear high dimension functions and benchmarks for the 0 1 multidimensional knapsack problem mkp are employed to evaluate their performances their results are compared with some well known modified binary pso and binary gravitational search algorithm bgsa the experimental results showed that the proposed methods improve the performance of bpso in terms of convergence speed and solution accuracy

an exponential time 2 approximation algorithm for bandwidth the bandwidth of a graph g on n vertices is the minimum b such that the vertices of g can be labeled from 1 to n such that the labels of every pair of adjacent vertices differ by at most b in this paper we present a 2 approximation algorithm for the bandwidth problem that takes worst case o 1 9797 n o 3 0 6 2 1 7 n time and uses polynomial space this improves both the previous best 2 and 3 approximation algorithms of cygan et al which have o 3 n and o 2 n worst case running time bounds respectively our algorithm is based on constructing bucket decompositions of the input graph a bucket decomposition partitions the vertex set of a graph into ordered sets called buckets of almost equal sizes such that all edges are either incident to vertices in the same bucket or to vertices in two consecutive buckets the idea is to find the smallest bucket size for which there exists a bucket decomposition the algorithm uses a divide and conquer strategy along with dynamic programming to achieve the improved time bound

a graphical analysis of ratio scaled paired comparison data one of the building blocks of the analytic hierarchy process saaty saaty t l 1977 a scaling method for priorities in hierarchical structures math psych 15 234 281 is the ratio scaled assessment of an agent s preferences between pairs of alternatives this article shows how it is possible to visualize such data and to detect cardinal and ordinal inconsistencies in the respondent s judgments the graphical method used to display preferences is not new in itself it dates back at least to the work of gower gower j c 1977 the analysis of asymmetry and orthogonality j r barra f brodeau g romier b van cutsem eds recent developments in statistics north holland amsterdam 109 123 and many variants of his technique can be found in the literature but gower plots provide an especially valuable diagnostic tool in the context of ahp as discussed and abundantly illustrated herein

a note on perron vectors for almost regular tournament matrices abstract let t be an almost regular tournament matrix of order n with right perron vector w we show that if the i th row sum of t is n 2 2 and the j th row sum is n 2 then w i w j thus in the round robin competition corresponding to t the ranking schemes of kendall and wei and of ramanujacharyula agree with the ranking generated by the row sums of t

capso centripetal accelerated particle swarm optimization meta heuristic search algorithms are developed to solve optimization problems such algorithms are appropriate for global searches because of their global exploration and local exploitation abilities swarm intelligence si algorithms comprise a branch of meta heuristic algorithms that imitate the behavior of insects birds fishes and other natural phenomena to find solutions for complex optimization problems in this study an improved particle swarm optimization pso scheme combined with newton s laws of motion the centripetal accelerated particle swarm optimization capso scheme is introduced capso accelerates the learning and convergence of optimization problems in addition the binary mode of the proposed algorithm binary centripetal accelerated particle swarm optimization bcapso is introduced for binary search spaces these algorithms are evaluated using nonlinear benchmark functions and the results are compared with the gravitational search algorithm gsa and pso in both the real and the binary search spaces moreover the performance of capso in solving the functions is compared with some well known pso algorithms in the literature the experimental results showed that the proposed methods enhance the performance of pso in terms of convergence speed solution accuracy and global optimality

evolutionary multimodal optimization using the principle of locality the principle of locality is one of the most widely used concepts in designing computing systems to explore the principle in evolutionary computation crowding differential evolution is incorporated with locality for multimodal optimization instead of generating trial vectors randomly the first method proposed takes advantage of spatial locality to generate trial vectors temporal locality is also adopted to help generate offspring in the second method proposed temporal and spatial locality are then applied together in the third method proposed numerical experiments are conducted to compare the proposed methods with the state of the art methods on benchmark functions experimental analysis is undertaken to observe the effect of locality and the synergy between temporal locality and spatial locality further experiments are also conducted on two application problems one is the varied line spacing holographic grating design problem while the other is the protein structure prediction problem the numerical results demonstrate the effectiveness of the methods proposed

eigenvalues of oriented graph matrices abstract we give bounds on the real and imaginary parts of the eigenvalues of an oriented graph matrix prove that each of the irreducible oriented graph matrices of order n 3 has at least three distinct eigenvalues and determine such irreducible matrices with three distinct eigenvalues we also consider spectral properties of the bipartite oriented graph matrices and the multiequipartite oriented graph matrices

solution of the conjecture of brualdi and li abstract we establish the conjecture of brualdi and li on the maximal perron root of a tournament matrix of even order

tournament matrices with extremal spectral properties abstract for a tournament matrix m of order n we define its walk space wm to be span m j 1 j 0 n 1 where 1 is the all ones vector we show that the dimension of wm equals the number of eigenvalues of m whose real parts are greater than 1 2 we then focus on tournament matrices whose walk space has particularly simple structure and characterize them in terms of their spectra specifically we characterize those tournament matrices such that mj1 is an eigenvector of m for some j 0 we also characterize the tournament matrices m such that jn 2m is a skew hadamard matrix throughout we illustrate our results with examples

pick s inequality and tournaments abstract let t be a tournament on n nodes and let a be its adjacency matrix a brauer and i gentry observed that an inequality due to g pick implies that im 1 2 cot 2n for all eigenvalues of a we say that t is a pick or p tournament if equality holds for some we determine when equality holds in pick s inequality for arbitrary real matrices and use this to show that the p tournaments can all be constructed from the transitive tournament m by reversing the arcs between the sets of certain node partitions or cuts u the cuts are specified by 1 n vectors u such that utw 0 where w 1 n 1 t e i n this links the cuts to cyclotomic polynomials there is at least one p tournament on n nodes if and only if n 2k k 1 up to isomorphism there is precisely one p tournament on n nodes if and only if n p or n 2p for some odd prime p for odd n up toisomorphism the only regular p tournament on n nodes has matrix zn circ 0 1 1 0 0 a composition rule is used on the matrices zp to form all p tournaments on n 2kpl nodes l 1