on kostant s theorem for the lie superalgebra q n abstract in this paper we study finite w algebras for basic superalgebras and q n associated with the regular even nilpotent coadjoint orbits we prove that this algebra satisfies the amitsur levitzki identity and therefore all its irreducible representations are finite dimensional in the case of q n we give an explicit description of the w algebra in terms of generators and relations and realize it as a quotient of the super yangian of q 1

applications of field theoretic renormalization group methods to reaction diffusion problems we review the application of field theoretic renormalization group rg methods to the study of fluctuations in reaction diffusion problems we first investigate the physical origin of universality in these systems before comparing rg methods to other available analytic techniques including exact solutions and smoluchowski type approximations starting from the microscopic reaction diffusion master equation we then pedagogically detail the mapping to a field theory for the single species reaction ka a k we employ this particularly simple but non trivial system to introduce the field theoretic rg tools including the diagrammatic perturbation expansion renormalization and callan symanzik rg flow equation we demonstrate how these techniques permit the calculation of universal quantities such as density decay exponents and amplitudes via perturbative dc d expansions with respect to the upper critical dimension dc with these basics established we then provide an overview of more sophisticated applications to multiple species reactions disorder effects l evy flights persistence problems and the influence of spatial boundaries we also analyse field theoretic approaches to non equilibrium phase transitions separating active from absorbing states we focus particularly on the generic directed percolation universality class as well as on the most prominent exception to this class even offspring branching and annihilating random walks finally we summarize the state of the field and present our perspective on outstanding problems for the future

horizontal visibility graphs exact results for random time series networks this procedure allows us to apply methods of complex network theory for characterizing time series in this work we present the horizontal visibility algorithm a geometrically simpler and analytically solvable version of our former algorithm focusing on the mapping of random series series of independent identically distributed random variables after presenting some properties of the algorithm we present exact results on the topological properties of graphs associated with random series namely the degree distribution the clustering coefficient and the mean path length we show that the horizontal visibility algorithm stands as a simple method to discriminate randomness in time series since any random series maps to a graph with an exponential degree distribution of the shape pk 1 32 3 k 2 independent of the probability distribution from which the series was generated accordingly visibility graphs with other pk are related to nonrandom series numerical simulations confirm the accuracy of the theorems for finite series in a second part we show that the method is able to distinguish chaotic series from independent and identically distributed i i d theory studying the following situations i noise free low dimensional chaotic series ii low dimensional noisy chaotic series even in the presence of large amounts of noise and iii high dimensional chaotic series coupled map lattice without needs for additional techniques such as surrogate data or noise reduction methods finally heuristic arguments are given to explain the topological properties of chaotic series and several sequences that are conjectured to be random are analyzed

superdiffusive transport due to random velocity fields the motion of a random walk in a medium containing random but spatially correlated velocity fields is discussed this type of disorder generally leads to superdiffusive behavior in which the mean square displacement x2 t grows faster than linearly with time for a two dimensional medium with layers of random velocities in the x direction x2 t is found to increase as t2v with 2v 32 the probability distribution of displacements appears to fit the form p x t t 34 exp x t34 with 1 7 however a lifshitz argument for the tail of the distribution suggests that 43 this discrepancy is yet to be fully resolved we also discuss some intriguing properties in a model with isotropic velocity fields in two dimensions we find v 43 while 3 these values obey the general scaling relation 1 v 1

duality for ext groups and extensions of discrete series for graded hecke algebras in this paper we study extensions of graded affine hecke algebra modules in particular based on an explicit projective resolution on graded affine hecke algebra modules we prove a duality result for ext groups this duality result with an ind res resolution gives an algebraic proof of the fact that all higher ext groups between a tempered module and a discrete series vanish

on the finite w algebra for the lie superalgebra q n in the non regular case in this paper we study the finite w algebra for the queer lie superalgebra q n associated with the non regular even nilpotent coadjoint orbits in the case when n nl and the corresponding nilpotent element has jordan blocks each of size l we prove that this finite w algebra is isomorphic to a quotient of the super yangian of q n

a unification of knizhnik zamolodchikov and dunkl operators via affine hecke algebras some generalizations of the lusztig lascoux schutzenberger operators for affine hecke algebras are considered as corollaries we obtain lusztig s isomorphisms from affine hecke algebras to their degenerate versions a natural interpretation of the dunkl operators and a new class of differential difference operators generalizing dunkl s ones and the knizhnik zamolodchikov operators from the two dimensional conformal field theory

regular articlethe structure of the young symmetrizers for spin representations of the symmetric group ii the first paper in this series established that the projective analogue of the young symmetrizer recently introduced by nazarov has a naturalpxq structure comparable with thepq form of the classical symmetrizer this second paper develops the theory on this decomposition further a more efficient construction of the projective symmetrizer is presented for certain classes of partitions this approach yields a compact expression for the intermediate factorx in the projective decomposition

relations between young s natural and the kazhdan lusztig representations of sn abstract our main result here is that under a suitable order of standard tableaux the classical representation of sn introduced by young in the collected papers of alfred young 1873 1940 univ of toronto press toronto qsa iv and usually referred as the natural representattion the the more recently discovered invent math 53 1979 165 184 kazhdan lusztig k l representation are related by an upper triangular integral matrix with unit diagonal elements we have been led to this discovery by a numerical exploration we noted it in each of the irreducible representations of sn up to n 6 the calculations in these cases were carried out by constructing the corresponding kazhdan lusztig graphs from tables m goresky tables of kazhdan lusztig polynomials unpublished of k l polynomials to extend the calculations to n 7 we have used graphs obtained by means of an algorithm given by lascoux and schutzenberger polynomes de kazhdan lusztig pour les grassmanniennes preprint remarkably the same property holds also for these graphs these findings appear to confirm the assertion made by these authors that their algorithm does indeed yield k l graphs for the case of hook shapes we have obtained an explicit construction of the transforming matrices a result which was also suggested by our numerical data for general shapes the transforming matrices are less explicit and our proof is based on certain properties of the kazhdan lusztig representations given in their article invent math 53 1979 165 184 and on a purely combinatorial construction of the natural representation

projective representations of symmetric groups via sergeev duality in this article we determine the irreducible projective representations of the symmetric group sd and the alternating group ad over an algebraically closed field of characteristic p 6 2 these matters are well understood in the case p 0 thanks to the fundamental work of schur 24 in 1911 as well as the much more recent work of nazarov 19 20 sergeev 25 26 and others so the focus here is primarily on the case of positive characteristic where surprisingly little is known as yet in particular we obtain a natural combinatorial labelling of the irreducibles in terms of a certain set rpp d of restricted p strict partitions of d such partitions arose recently in work of kashiwara miwa peterson and yung 11 and leclerc and thibon 14 on the q deformed fock space of the affine kac moody algebra of type a p 1 leclerc and thibon proposed that rpp d should label the irreducible projective representations in some natural way and we show here how this can be done note that for p 3 5 the labelling problem was solved in 1 3 while if p 2 all projective representations of sd and ad are linear so do not need to be considered further here to be more precise recall that is a partition of d if 1 2 is a non increasing sequence of non negative integers summing to d call p strict if in addition

affine sergeev algebra and q analogues of the young symmetrizers for projective representations of the symmetric group we study a q deformation for the crossed product of the symmetric group s n with the clifford algebra on n anticommuting generators we obtain a q version of the projective analogue for the classical young symmetrizer introduced by the second author our main tool is an analogue of the hecke algebra of complex valued functions on the group mbox gl n over a p adic field relative to the iwahori subgroup

a characterization of horizontal visibility graphs and combinatorics on words abstract a horizontal visibility graph hvg is defined in association with an ordered set of non negative reals hvgs realize a methodology in the analysis of time series their degree distribution being a good discriminator between randomness and chaos luque et al b luque l lacasa f ballesteros j luque horizontal visibility graphs exact results for random time series phys rev e 80 2009 046103 we prove that a graph is an hvg if and only if it is outerplanar and has a hamilton path therefore an hvg is a noncrossing graph as defined in algebraic combinatorics flajolet and noy p flajolet m noy analytic combinatorics of noncrossing configurations discrete math 204 1999 203 229 our characterization of hvgs implies a linear time recognition algorithm treating ordered sets as words we characterize subfamilies of hvgs highlighting various connections with combinatorial statistics and introducing the notion of a visible pair with this technique we determine asymptotically the average number of edges of hvgs

extensions and applications of equitable decompositions for graphs with symmetries abstract we extend the theory of equitable decompositions introduced in 2 where it was shown that if a graph has a particular type of symmetry i e a uniform or basic automorphism it is possible to use to decompose a matrix m appropriately associated with the graph the result is a number of strictly smaller matrices whose collective eigenvalues are the same as the eigenvalues of the original matrix m we show here that a large class of automorphisms which we refer to as separable can be realized as a sequence of basic automorphisms allowing us to equitably decompose m over any such automorphism we also show that not only can a matrix m be decomposed but that the eigenvectors of m can also be equitably decomposed additionally we prove under mild conditions that if a matrix m is equitably decomposed the resulting divisor matrix which is the divisor matrix of the associated equitable partition will have the same spectral radius as the original matrix m last we describe how an equitable decomposition effects the gershgorin region m of a matrix m which can be used to localize the eigenvalues of m we show that the gershgorin region of an equitable decomposition of m is contained in the gershgorin region m of the original matrix we demonstrate on a real world network that by a sequence of equitable decompositions it is possible to significantly reduce the size of a matrix gershgorin region

anomalous diffusion in a unidirectional random velocity field with long range correlations the anomalous diffusion in a unidirectional random velocity field with long range correlations is analysed in directions transverse to the direction of the field or along the field direction critical values of the exponents which characterize the power like falloff of the correlations in the transverse and longitudinal directions and the critical dimension of space are determined the anomalous dimension of the longitudinal diffusion coefficient is also calculated in the first order of the in expansion for several cases of a long range correlated random velocity field

sklyanin determinant laplace operators and characteristic identities for classical lie algebras the structure of quantized enveloping algebras called twisted yangians which are naturally associated with the b c and d series of the classical lie algebras is studied an explicit formula for the formal series the sklyanin determinant whose coefficients are free generators of the center of the twisted yangian is obtained as a corollary a new system of algebraically independent generators of the center of the universal enveloping algebra for the orthogonal and symplectic lie algebras is obtained and the characteristic polynomial for the matrix formed by the generators of these lie algebras is found

spin hecke algebras of finite and affine types we introduce the spin hecke algebra which is a q deformation of the spin symmetric group algebra and its affine generalization we establish an algebra isomorphism which relates our spin affine hecke algebras to the affine hecke clifford algebras of olshanski and jones nazarov relation between the spin affine hecke algebra and a nonstandard presentation of the usual affine hecke algebra is displayed and the notion of covering affine hecke algebra is introduced to provide a link between these algebras various algebraic structures for the spin affine hecke algebra are established

twisted affine lie superalgebra of type q and quantization of its enveloping superalgebra we introduce a new quantum group which is a quantization of the enveloping superalgebra of a twisted affine lie superalgebra of type q we study generators and relations for superalgebras in the finite and twisted affine cases and also universal central extensions afterwards we apply the frt formalism to a certain solution of the quantum yang baxter equation to define that new quantum group and we study some of its properties we construct a functor of schur weyl type which connects it to affine hecke clifford algebras and prove that it provides an equivalence between two categories of modules

spin nilhecke algebras of classical type abstract we formulate and study the spin nilhecke algebras nh n b and nh n d of type b d which differ from the usual nilhecke algebras by some odd signs the type b spin nilhecke algebra is a nil version of the spin type b hecke algebra introduced earlier by the second author and khongsap but not for the type d one we construct faithful polynomial representations pol n of the nilhecke algebras via odd demazure operators we formulate the spin schubert polynomials and use them to show that the spin nilhecke algebras are matrix algebras with entries in a subalgebra of pol n consisting of spin symmetric polynomials all these results have their counterparts for the usual nilhecke algebras over the rational field our work is a generalization of results of lauda and ellis khovanov lauda in usual spin type a

renormalization group analysis of superdiffusion in random velocity fields field theoretic renormalization group theory is applied to the analysis of the asymptotic behaviour of diffusion in random velocity fields each nonvanishing component of which is independent of the coordinate along the direction of that component below the critical dimension dc 3 the corresponding field theory is shown to have a nontrivial infrared stable fixed point of the renormalization group which controls the long time large distance behaviour of the model the anomalous dimension nu has been found exactly in the epsilon 3 d expansion and the leading logarithmic corrections to the normal diffusion at d 3 have been determined for all d 1 or d or d where d is the number of components of the drift field

the projective representations of the hyperoctahedral group the hyperoctahedral group the weyl group of the root system bn has seven distinct nonsplit double covers for n 4 and hence seven families of projective representations we give constructions of all the irreducible representations of these seven double covers in terms of symmetric group representations and determine the associated character tables as a corollary we also obtain the irreducible projective representations and characters of the weyl group of the root system dn

superdiffusion in random velocity fields abstract stochastic transport in a medium containing random but spatially correlated velocity fields is discussed this type of disorder generally leads to superdiffusive behavior in which the mean square displacement of a random walk x2 t grows faster than linearly with time for a two dimentional layered medium with y dependent random velocities in the x direction ux y x2 x t2 with v 3 4 and with strong sample to sample fluctuations the probability distribution of displacements averaged over environments takes a non gaussian scaling form at large time p x t 3 4 x t 3 4 where u exp u for u 1 with 4 3 for an isotropic two dimensional medium with ux y having the same statistical properties we find v 2 3 and 1 1 3 for the layered medium the moments of the time for a random walker to first reach a distance x in the longitudinal direction increases as t n 1 n x 4 3 possibly modified by logarithmic corrections however the probability that the walk has not reached a distance x in a time t decreases asymptotically as e t with x2 indicating that more than a single time scale is needed to account for first passage properties

finite dimensional irreducible representations of twisted yangians we study quantized enveloping algebras called twisted yangians they are analogs of the yangian y gl n for the classical lie algebras of b c and d series the twisted yangians are subalgebras in y gl n and coideals with respect to the coproduct in y gl n we give a complete description of their finite dimensional irreducible representations every such representation is highest weight and we give necessary and sufficient conditions for an irreducible highest weight representation to be finite dimensional the result is analogous to drinfeld s theorem for the ordinary yangians its detailed proof for the a series is also reproduced for the simplest twisted yangians we construct an explicit realization for each finite dimensional irreducible representation in tensor products of representations of the corresponding lie algebras

yangian of the queer lie superalgebra consider the complex matrix lie superalgebra with the standard generators where define an involutory automorphism of by the twisted polynomial current lie superalgebra has a natural lie co superalgebra structure we quantise the universal enveloping algebra as a co poisson hopf superalgebra for the quantised algebra we give a description of the centre and construct the double in the sense of drinfeld we also construct a wide class of irreducible representations of the quantised algebra

representations of super yangian the classification of the finite dimensional irreducible representations of the super yangian associated with the lie superalgebra gl 1 1 is presented in detail

a categorification of twisted heisenberg algebras abstract we categorify a quantized twisted heisenberg algebra associated to a finite subgroup of sl 2 c

clifford algebra as quantum language we suggest clifford algebra as a useful simplifying language for present quantum dynamics clifford algebras arise from representations of the permutation groups as they arise from representations of the rotation groups aggregates using such representations for their permutations obey clifford statistics the vectors supporting the clifford algebras of permutations and rotations are plexors and spinors respectively physical spinors may actually be plexors describing quantum ensembles not simple individuals we use clifford statistics to define quantum fields on a quantum space time and to formulate a quantum dynamics field space time unity that evades the compactification problem the quantum bits of history regarded as a quantum computation seem to obey a clifford statistics

the structure of the young symmetrizers for spin representations of the symmetric group i abstract this paper is the first in a series of three papers on the young symmetrizers for the spin representations of the symmetric group in this opening paper it is shown that the projective analogue of the young symmetrizer recently introduced by nazarov has a structure resembling thep q form exhibited by the classical symmetrizer in contrast to the classical case our decomposition of the projective symmetrizer requires an additional intermediate factorx this factor will be studied in greater detail in the later papers

visibility graphs and symbolic dynamics abstract visibility algorithms are a family of geometric and ordering criteria by which a real valued time series of n data is mapped into a graph of n nodes this graph has been shown to often inherit in its topology nontrivial properties of the series structure and can thus be seen as a combinatorial representation of a dynamical system here we explore in some detail the relation between visibility graphs and symbolic dynamics to do that we consider the degree sequence of horizontal visibility graphs generated by the one parameter logistic map for a range of values of the parameter for which the map shows chaotic behaviour numerically we observe that in the chaotic region the block entropies of these sequences systematically converge to the lyapunov exponent of the time series hence pesin s identity suggests that these block entropies are converging to the kolmogorov sinai entropy of the physical measure which ultimately suggests that the algorithm is implicitly and adaptively constructing phase space partitions which might have the generating property to give analytical insight we explore the relation k x x 0 1 that for a given datum with value x assigns in graph space a node with degree k in the case of the out degree sequence such relation is indeed a piece wise constant function by making use of explicit methods and tools from symbolic dynamics we are able to analytically show that the algorithm indeed performs an effective partition of the phase space and that such partition is naturally expressed as a countable union of subintervals where the endpoints of each subinterval are related to the fixed point structure of the iterates of the map and the subinterval enumeration is associated with particular ordering structures that we called motifs

new boundary conditions for integrable lattices new boundary conditions for classical integrable nonlinear lattices of the xxx type such as the heisenberg chain and the toda lattice are presented these integrable extensions are formulated in terms of a generic xxx heisenberg magnet interacting with two additional spins at each end of the chain the construction uses the most general rank 1 ansatz for the 2 2 l operator satisfying the reflection equation algebra with rational r matrix the associated quadratic algebra is shown to be that of dynamical symmetry for the a1 and bc2 calogero moser problems other physical realizations of our quadratic algebra are also considered

cocenters of hecke clifford and spin hecke algebras abstract we determine a basis of the cocenter i e the trace or zeroth hochschild homology of the degenerate affine hecke clifford and spin hecke algebras in classical types

the orthogonal eigenbasis and norms of eigenvectors in the spin calogero sutherland model using a technique based on the yangian gelfand zetlin algebra and the associated yangian gelfand zetlin bases we construct an orthogonal basis of eigenvectors in the calogero sutherland model with spin and derive product type formulae for norms of these eigenvectors

twisted yangians twisted quantum loop algebras and affine hecke algebras of type bc we study twisted yangians of type aiii which have appeared in the literature under the name of reflection algebras they admit q versions which are new twisted quantum loop algebras we explain how these can be defined equivalently either via the reflection equation or as coideal subalgebras of yangians of gln resp of quantum loop algebras of gln the connection with affine hecke algebras of type bc comes from a functor of schur weyl type between their module categories

algebraic structures on grothendieck groups of a tower of algebras abstract the grothendieck group of the tower of symmetric group algebras has a self dual graded hopf algebra structure inspired by this we introduce by way of axioms a general notion of a tower of algebras and study two grothendieck groups on this tower linked by a natural paring using representation theory we show that our axioms give a structure of graded hopf algebras on each grothendieck groups and these structures are dual to each other we give some examples to indicate why these axioms are necessary we also give auxiliary results that are helpful to verify the axioms we conclude with some remarks on generalized towers of algebras leading to a structure of generalized bialgebras in the sense of loday on their grothendieck groups

degenerate affine hecke algebras and centralizer construction for the symmetric groups in our recent papers the centralizer construction was applied to the series of classical lie algebras to produce the quantum algebras called twisted yangians here we extend this construction to the series of the symmetric groups s n we study the stable properties of the centralizers of s n m in the group algebra c s n as n with m fixed we construct a limit centralizer algebra a and describe its algebraic structure the algebra a turns out to be closely related with the degenerate affine hecke algebras we also show that the so called tame representations of s yield a class of natural a modules

yang baxter equation in long range interacting systems we consider the su p spin chains with long range interactions and the spin generalization of the calogero sutherland models we show that their properties derive from a transfer matrix obeying the yang baxter equation we obtain the expression of the conserved quantities of the dynamical models and we diagonalise them in the spin chain case we establish the connection between the degeneracies of the spectrum and the representation theory of the yangians we use a correspondence with the dynamical models to diagonalise the hamiltonian finally we extend the previous results to the case of a trigonometric r matrix

type a blocks of super category o we show that every block of category oo for the general linear lie superalgebra glm n k glm n k is equivalent to some corresponding block of category oo for the queer lie superalgebra qm n k qm n k this implies the truth of the kazhdan lusztig conjecture for the so called type a blocks of category oo for the queer lie superalgebra as formulated by cheng kwon and wang

reflection equation and twisted yangians with any involutive antialgebra and coalgebra automorphism of a quasitriangular bialgebra we associate a reflection equation algebra a hopf algebraic treatment of the reflection equation of this type and its universal solution is given applications to the twisted yangians are considered

quantum group symmetries and non local currents in 2d qft we construct and study the implications of some new non local conserved currents that exist is a wide variety of massive integrable quantum field theories in 2 dimensions including the sine gordon theory and its generalization to affine toda theory these non local currents provide a non perturbative formulation of the theories the symmetry algebras correspond to the quantum affine kac moody algebras thes matrices are completely characterized by these symmetries formals matrices for the imaginary coupling affine toda theories are thereby derived the application of theses matrices to perturbed coset conformal field theory is studied non local charges generating the finite dimensional quantum group in the liouville theory are briefly presented the formalism based on non local charges we describe provides an algernative to the quantum inverse scattering method for solving integrable quantum field theories in 2d

bethe subalgebras in twisted yangians we study analogues of the yangian of the lie algebra mathfrak g mathfrak l n for the other classical lie algebras mathfrak s mathfrak o n and mathfrak s mathfrak p n we call them twisted yangians they are coideal subalgebras in the yangian of mathfrak g mathfrak l n and admit homomorphisms onto the universal enveloping algebras u mathfrak s mathfrak o n and u mathfrak s mathfrak p n respectively in every twisted yangian we construct a family of maximal commutative subalgebras parametrized by the regular semisimple elements of the corresponding classical lie algebra the images in u mathfrak s mathfrak o n and u mathfrak s mathfrak p n of these subalgebras are also maximal commutative

center of the quantum affine vertex algebra in type a abstract we consider the quantum vertex algebra associated with the double yangian in type a as defined by etingof and kazhdan we show that its center is a commutative associative algebra and construct algebraically independent families of topological generators of the center at the critical level

algebraic structures related to reflection equations quadratic algebras related to the reflection equations are introduced they are quantum group comodule algebras the quantum group fq gl 2 is taken as the example the properties of the algebras centre representation realizations real forms fusion procedure etc as well as the generalizations are discussed

equivariant k theory generalized symmetric products and twisted heisenberg algebra for a space x acted on by a finite group the product space xn affords a natural action of the wreath product n n sn the direct sum of equivariant k groups were shown earlier by the author to carry several interesting algebraic structures in this paper we study the k groups of n equivariant clifford supermodules on xn we show that is a hopf algebra and it is isomorphic to the fock space of a twisted heisenberg algebra twisted vertex operators make a natural appearance the algebraic structures on x when is trivial and x is a point specialize to those on a ring of symmetric functions with the schur q functions as a linear basis as a by product we present a novel construction of k theory operations using the spin representations of the hyperoctahedral groups

shifted tableaux and the projective representations of symmetric groups les groupes de representation de s n classes de conjugaison algebres de clifford et la representation spin de base produits induits pour des representations spin une application caracteristique pour des caracteres de spin tableaux decales et q fonctions de schur les caracteres de spin irreductibles de s n et a n une analogue decalee de la regle lr produits tensoriels interieurs

hecke clifford algebras and spin hecke algebras iii the trigonometric type abstract the notions of trigonometric spin double affine hecke algebras tsdaha and trigonometric double affine hecke clifford algebras tdahca associated to classical weyl groups are introduced the pbw basis property is established an algebra isomorphism relating tdahca to tsdaha is obtained

what is a complex graph abstract many papers published in recent years show that real world graphs g n m n nodes m edges are more or less complex in the sense that different topological features deviate from random graphs here we narrow the definition of graph complexity and argue that a complex graph contains many different subgraphs we present different measures that quantify this complexity for instance c 1 e the relative number of non isomorphic one edge deleted subgraphs i e deck size however because these different subgraph measures are computationally demanding we also study simpler complexity measures focussing on slightly different aspects of graph complexity we consider heuristically defined product measures the products of two quantities which are zero in the extreme cases of a path and clique and entropy measures quantifying the diversity of different topological features the previously defined network graph complexity measures medium articulation and offdiagonal complexity odc belong to these two classes we study odc measures in some detail and compare it with our new measures for all measures the most complex graph g c max has a medium number of edges between the edge numbers of the minimum and the maximum connected graph n 1 m g c max n n 1 2 interestingly for some measures c this number scales exactly with the geometric mean of the extremes m g c max n 2 n 1 n 1 5 all graph complexity measures are characterized with the help of different example graphs for all measures the corresponding time complexity is given finally we discuss the complexity of 33 real world graphs of different biological social and economic systems with the six computationally most simple measures including odc the complexities of the real graphs are compared with average complexities of two different random graph versions complete random graphs just fixed n m and rewired graphs with fixed node degrees