some extremal properties of the resolvent energy estrada and resolvent estrada indices of graphs abstract let g be a simple connected graph on n vertices and 1 2 n be the eigenvalues of the adjacency matrix of g estrada and higham proposed an invariant of a graph g based on taylor series expansion of spectral moments e e g c k 0 c k m k g for c k 1 n k resp 1 k 1 n 1 k e e g c is the resolvent energy resp estrada index resolvent estrada index of g in 18 17 gutman et al conjectured the structure of the extremal members of some classes of graphs by the aid of computer on resolvent energy and resolvent estrada index respectively in 1 l allem et al confirmed the validity of some of these conjectures on resolvent energy in this paper we continue to study these indices based on these conjectures

extremal estrada indices of the weighted trees with fixed total weight sum abstract let t n m be the set of positively weighted trees of order n with the fixed total weight sum m where all the weights are integers and m n 1 3 among t n m the weighted trees with the maximal and the second maximal estrada indices are derived for m n 1 3 and m n 1 7 respectively

anti modularity and anti community detecting in complex networks abstract many networks of interest in sciences and social research can be divided naturally into anti communities the problem of detecting and characterizing such anti community structure has attracted recent attention in this paper we first define the anti modularity as a quantitative measure of anti community partitioning on a network we also theoretically and empirically show the reliability of anti modularity as a measurement of the quality of an anti community partitioning a label propagation algorithm lpad for anti community detection is proposed experimental results on synthetic and real world networks show that our algorithm lpad can obtain higher quality anti community partitioning than other methods

subgraph centrality in complex networks we introduce a new centrality measure that characterizes the participation of each node in all subgraphs in a network smaller subgraphs are given more weight than larger ones which makes this measure appropriate for characterizing network motifs we show that the subgraph centrality c s i can be obtained mathematically from the spectra of the adjacency matrix of the network this measure is better able to discriminate the nodes of a network than alternate measures such as degree closeness betweenness and eigenvector centralities we study eight real world networks for which c s i displays useful and desirable properties such as clear ranking of nodes and scale free characteristics compared with the number of links per node the ranking introduced by c s i for the nodes in the protein interaction network of s cereviciae is more highly correlated with the lethality of individual proteins removed from the proteome

perturbation results for the estrada index in weighted networks the logarithm of the estrada index has been proposed recently as a spectral measure to character efficiently the robustness of complex networks in this paper we explore the estrada index in weighted networks and develop various perturbation results based on spectral graph theory it is shown that the robustness of a network may be enhanced even when some edge weights are reduced this is of particular theoretical and practical significance to network design and optimization

new topologies on colombeau generalized numbers and the fermat reyes theorem abstract based on the theory of fermat reals we introduce new topologies on spaces of colombeau generalized points and derive some of their fundamental properties in particular we obtain metric topologies on the space of near standard generalized points that induce the standard euclidean topology on the reals we analyze the description of the sharp topology in terms of the natural extension of the absolute value or of the defining semi norms in the case of locally convex spaces thereby preserving a number of classical notions building on a new point value characterization of colombeau generalized functions we prove a fermat reyes theorem that forms the basis of an approach to differentiation on spaces of generalized functions close to the classical one

fronts pulses sources and sinks in generalized complex ginzberg landau equations an important clement in the long time dynamics of pattern forming systems is a class of solutions we will call coherent structures these are states that are either themselves localized or that consist of domains of regular patterns connected by localized defects or interfaces this paper summarizes and extends recent work on such coherent structures in the one dimensional complex ginzburg landau equation and its generalizations for which rather complete information can be obtained on the existence and competition of fronts pulses sources and sinks for the special subclass of uniformly translating structures the solutions are derived from a set of ordinary differential equations that can be interpreted as a flow in a three dimensional phase space fixed points of the flow correspond to the two basic building blocks of coherent structures uniform amplitude states and evanescent waves whose amplitude decreases smoothly to zero a study of the stability of the fixed points under the flow leads to results on the existence and multiplicity of the different coherent structures the dynamical analysis of the original partial differential equation focusses on the competition between pulses and fronts and is expressed in terms of a set of conjectures for front propagation that generalize the marginal stability and pinch point approaches of earlier authors these rules together with an exact front solution whose dynamics plays an important role in the selection of patterns yield an analytic expression for the upper limit of the range of existence of pulse solutions as well as a determination of the regions of parameter space where uniformly translating fron t solutions can exist extensive numerical simulations show consistency with these rules and conjectures for the existence of fronts and pulses in the parameter ranges where no uniformly translating fronts can exist examples are shown of irregularly spreading fronts that generate strongly chaotic regions as well as nonuniformly translating fronts that lead to uniform amplitude states recent perturbative treatments based on expansions about the nonlinear schr6dinger equation are generalized to perturbations of the cubic quintic and derivative schr idinger equations for which both pulses and fronts exist in the unperturbed system comparison of the results with the exact solutions shows that the perturbation theory only yields a subset of the relevant solutions nevertheless those that are obtained are found to be consistent with the general conjectures and in particular they provide an analytic demonstration of front pulse competition while the discussion of the competition between fronts and pulses focusses on the complex ginzburg landau equation with quintic terms and a subcritical bifurcation a number of results are also presented for the cubic equation in particular the existence of a family of moving source solutions derived by bekki and nozaki for this equation contradicts the naive counting arguments we attribute this contradiction to a hidden symmetry of the solution but have not been able to show explicitly how this symmetry affects the phase space orbits

nonstandard principles for generalized functions abstract we show that principles from nonstandard analysis hold to some extent for nonlinear generalized functions the generalized functions under consideration are constructed as families of functions modulo a free filter as it is usually done in applied analysis in contrast with models of nonstandard analysis we do not require the filter to be an ultrafilter the principles are intended to be used as a tool for proving theorems which we illustrate by means of an automatic continuity result that was not suspected by experts in the field

finding community structure in networks using the eigenvectors of matrices we consider the problem of detecting communities or modules in networks groups of vertices with a higher than average density of edges connecting them previous work indicates that a robust approach to this problem is the maximization of the benefit function known as modularity over possible divisions of a network here we show that this maximization process can be written in terms of the eigenspectrum of a matrix we call the modularity matrix which plays a role in community detection similar to that played by the graph laplacian in graph partitioning calculations this result leads us to a number of possible algorithms for detecting community structure as well as several other results including a spectral measure of bipartite structure in networks and a centrality measure that identifies vertices that occupy central positions within the communities to which they belong the algorithms and measures proposed are illustrated with applications to a variety of real world complex networks

gaussianization of the spectra of graphs and networks theory and applications abstract matrix functions of the adjacency matrix are very useful for understanding important structural properties of graphs and networks such as communicability node centrality bipartivity and many more they are also intimately related to the solution of differential equations describing dynamical processes on graphs and networks here we propose a new matrix function based on the gaussianization of the adjacency matrix of a graph this function gives more weight to a selected reference eigenvalue ref which may be located in any region of the graph spectra we show here that this matrix function can be derived from physical models that consider the interactions between nearest and next nearest neighbors in the graph we first obtain a few mathematical results for the trace of this matrix function when ref 1 h 1 for simple graphs as well as for random graphs we also provide a combinatorial interpretation of this index in terms of subgraphs in the graph and in terms of the competition pressure among agents in a complex system finally we apply this index to the study of magnetic properties of molecules emerging due to spin interactions as well as to studying the temporal evolution of the international trade network in the period 1992 2002 in both cases we give a clear phenomenological interpretation of the processes described

an increasing sequence of lower bounds for the estrada index of graphs and matrices abstract let g be a graph on n vertices and 1 2 n its eigenvalues the estrada index of g is defined as e e g i 1 n e i in this work we use an increasing sequence converging to the 1 to obtain an increasing sequence of lower bounds for e e g in addition we generalize this succession for the estrada index of an arbitrary symmetric nonnegative matrix

sources sinks and wavenumber selection in coupled cgl equations and experimental implications for counter propagating wave systems we study the coupled complex ginzburg landau cgl equations for traveling wave systems and show that sources and sinks are the important coherent structures that organize much of the dynamical properties of traveling wave systems we focus on the regime in which sources and sinks separate patches of left and right traveling waves i e the case that these modes suppress each other we present in detail the framework to analyze these coherent structures and show that the theory predicts a number of general properties which can be tested directly in experiments our counting arguments for the multiplicities of these structures show that independently of the precise values of the coefficients in the equations there generally exists a symmetric stationary source solution which sends out waves with a unique frequency and wave number sinks on the other hand occur in two parameter families and play an essentially passive role being sandwiched between the sources these simple but general results imply that sources are important in organizing the dynamics of the coupled cgl equations simulations show that the consequences of the wavenumber selection by the sources is reminiscent of a similar selection by spirals in the 2d complex ginzburg landau equations sources can send out stable waves convectively unstable waves or absolutely unstable waves we show that there exists an additional dynamical regime where both single and bimodal states are unstable the ensuing chaotic states have no counterpart in single amplitude equations a third dynamical mechanism is associated with the fact that the width of the sources does not show simple scaling with the growth rate e this is related to the fact that the standard coupled cgl equations are not uniform in e in particular when the group velocity term dominates over the linear growth term no stationary source can exist however sources displaying nontrivial dynamics can often survive here our results for the existence multiplicity wavelength selection dynamics and scaling of sources and sinks and the patterns they generate are easily accessible by experiments we therefore advocate a study of the sources and sinks as a means to probe traveling wave systems and compare theory and experiment in addition they bring up a large number of new research issues and open problems which are listed explicitly in the concluding section

understanding how both the partitions of a bipartite network affect its one mode projection abstract it is a well known fact that the degree distribution dd of the nodes in a partition of a bipartite network influences the dd of its one mode projection on that partition however there are no studies exploring the effect of the dd of the other partition on the one mode projection in this article we show that the dd of the other partition in fact has a very strong influence on the dd of the one mode projection we establish this fact by deriving the exact or approximate closed forms of the dd of the one mode projection through the application of generating function formalism followed by the method of iterative convolution the results are cross validated through appropriate simulations

a note on new bounds for the estrada index abstract let g be a graph on n vertices and 1 2 n their eigenvalues the estrada index of g is defined as e e g i 1 n e i in this paper we obtain new upper and lower bounds on the estrada index of a given graph considering numerical inequalities present in the literature

internal sets and internal functions in colombeau theory this paper is devoted to the study of generalized functions as pointwise functions so called internal functions on certain sets of generalized points so called internal sets we treat the case of the colombeau algebras of generalized functions for which these notions have turned out to constitute a fundamental technical tool we provide general foundations for the notion of internal functions and internal sets and prove a saturation principle various applications to colombeau algebras are given

front propagation into unstable states universal algebraic convergence towards uniformly translating pulled fronts fronts that start from a local perturbation and propagate into a linearly unstable state come in two classes pulled fronts and pushed fronts the term pulled front expresses that these fronts are pulled along by the spreading of linear perturbations about the unstable state accordingly their asymptotic speedv equals the spreading speed of perturbations whose dynamics is governed by the equations linearized about the unstable state the central result of this paper is the analysis of the convergence of asymptotically uniformly traveling pulled fronts towardsv we show that when such fronts evolve from sufficiently steep initial conditions which initially decay faster than e x forx 1 t hey have auniversal relaxation behavioras timet 1 the velocity of a pulled front always relaxes algebraically like v t d v 3 2t c 3 p d d 2 t 3 2 c o 1 t 2 the parametersv and d are determined through a saddle point analysis from the equation of motion linearized about the unstable invaded state this front velocity is independent of the precise value of the front amplitude which one tracks to measure the front position the interior of the front is essentially slaved to the leading edge and develops universally as x t d8v t x rt dt 0 v t 0 c o 1 t 2 where8v x vt is a uniformly translating front solution with velocity v v our result which can be viewed as a general center manifold result for pulled front propagation is derived in detail for the well known nonlinear diffusion equation of type t d 2 x c 3 where the invaded unstable state is d 0 even for this simple case the subdominant t 3 2 term extends an earlier result of bramson our analysis is then generalized to more general sets of partial differential equations with higher spatial or temporal derivatives to pdes with memory kernels and also to difference equations such as those that occur in numerical finite difference codes our universal result for pulled fronts thus implies independence i of the level curve which is used to track the front position ii of the precise nonlinearities iii of the precise form of the linear operators in the dynamical equation and iv of the precise initial conditions as long as they are sufficiently steep the only remainders of the explicit form of the dynamical equation are the nonlinear solutions 8v and the three saddle point parameters v and d as our simulations confirm all our analytical predictions in every detail it can be concluded that we have a complete analytical understanding of the propagation mechanism and relaxation behavior of pulled fronts if they are uniformly translating for t 1 an immediate consequence of the slow algebraic relaxation is that the standard moving boundary approximation breaks down for weakly curved pulled fronts in two or three dimensions in addition to our main result for pulled fronts we also discuss the propagation and convergence of fronts emerging from

spectral moments of trees with given degree sequence abstract let 1 n be the eigenvalues of a graph g for any k 0 the k th spectral moment of g is defined by m k g 1 k n k we use the fact that m k g is also the number of closed walks of length k in g to show that among trees t whose degree sequence is d or majorized by d m k t is maximized by the greedy tree with degree sequence d constructed by assigning the highest degree in d to the root the second third highest degrees to the neighbors of the root and so on for any k 0 several corollaries follow in particular a conjecture of ilic and stevanovic on trees with given maximum degree which in turn implies a conjecture of gutman furtula markovic and glisic on the estrada index of such trees which is defined as ee g e 1 e n

on the extended estrada index of some graphs abstract let g be a graph on n vertices and 1 2 n the eigenvalues of its extended adjacency matrix the extended estrada index eeex is defined as the sum of the terms e i i 1 2 n in this paper we show the extended estrada index of the complete bipartite graphs and the friendship graphs also the relationship of the extended estrada index between the m splitting m shadow graphs and the original graphs are obtained

on the estrada index conjecture abstract let g be a simple graph of order n with m edges let the adjacency spectrum be 1 2 n 1 n of g where 1 2 n 1 n the estrada index of a graph g is ee g i 1 n e i in j a pena i gutman j rada estimating the estrada index linear algebra appl 427 2007 70 76 pena et al posed a conjecture that the star s n has maximum estrada index for any tree of order n and the path p n has minimum estrada index for any tree of order n or any connected graph of order n in this paper we have proved that the star has maximum estrada index for any tree also we obtain that the path has minimum estrada index for any connected graph with m 1 8 n 4 or m n 2 6 moreover we give better lower bound on estrada index for any connected graph

spectral measures of bipartivity in complex networks we introduce a quantitative measure of network bipartivity as a proportion of even to total number of closed walks in the network spectral graph theory is used to quantify how close to bipartite a network is and the extent to which individual nodes and edges contribute to the global network bipartivity it is shown that the bipartivity characterizes the network structure and can be related to the efficiency of semantic or communication networks trophic interactions in food webs construction principles in metabolic networks or communities in social networks

traveling waves in diffusive predator prey equations periodic orbits and point to periodic heteroclinic orbits we demonstrate the existence of periodic traveling wave train and traveling front solutions for a diffusive predator prey system the analysis is in the three dimensional phase space of the nonlinear ordinary differential equation system given by the diffusive predator prey system in the traveling wave variable the analysis shows the existence of periodic orbits heteroclinic orbits and a heteroclinic connection of a point and a periodic orbit the proof uses shooting techniques invariant manifold theory and the qualitative theory of ordinary differential equations

on network bipartivity systems with two types of agents with a preference for heterophilous interaction produces networks that are more or less close to bipartite we propose two measures quantifying the notion of bipartivity the two measures one well known and natural but computationally intractable one computationally less complex but also less intuitive are examined on model networks that continuously interpolates between bipartite graphs and graphs with many odd circuits we find that the bipartivity measures increase as we tune the control parameters of the test networks to intuitively increase the bipartivity and thus conclude that the measures are quite relevant we also measure and discuss the values of our bipartivity measures for empirical social networks constructed from professional collaborations internet communities and field surveys here we find as expected that networks arising from romantic online interaction have high and professional collaboration networks have low bipartivity values in some other cases probably due to low average degree of the network the bipartivity measures cannot distinguish between romantic and friendship oriented interaction

communicability in complex networks we propose a new measure of the communicability of a complex network which is a broad generalization of the concept of the shortest path according to the new measure most of the real world networks display the largest communicability between the most connected popular nodes of the network assortative communicability there are also several networks with the disassortative communicability where the most popular nodes communicate very poorly to each other using this information we classify a diverse set of real world complex systems into a small number of universality classes based on their structure dynamic correlation in addition the new communicability measure is able to distinguish finer structures of networks such as communities into which a network is divided a community is unambiguously defined here as a set of nodes displaying larger communicability among them than to the rest of the nodes in the network

on maximum estrada indices of k trees abstract the estrada index of a graph g is defined as ee g i 1 n e i where 1 2 n are the eigenvalues of the adjacency matrix of g let t n k be the set of the k trees of order n in this paper we characterize the graphs among t n k having the first resp the second largest estrada index

graph animals subgraph sampling and motif search in large networks we generalize a sampling algorithm for lattice animals connected clusters on a regular lattice to a monte carlo algorithm for graph animals i e connected subgraphs in arbitrary networks as with the algorithm in n kashtan et al bioinformatics 20 1746 2004 it provides a weighted sample but the computation of the weights is much faster linear in the size of subgraphs instead of superexponential this allows subgraphs with up to ten or more nodes to be sampled with very high statistics from arbitrarily large networks using this together with a heuristic algorithm for rapidly classifying isomorphic graphs we present results for two protein interaction networks obtained using the tandem affinity purification tap method one of escherichia coli with 230 nodes and 695 links and one for yeast saccharomyces cerevisiae with roughly ten times more nodes and links we find in both cases that most connected subgraphs are strong motifs z scores 10 or antimotifs z scores 10 when the null model is the ensemble of networks with fixed degree sequence strong differences appear between the two networks with dominant motifs in e coli being nearly bipartite graphs and having many pairs of nodes that connect to the same neighbors while dominant motifs in yeast tend towards completeness or contain large cliques we also explore a number of methods that do not rely on measurements of z scores or comparisons with null models for instance we discuss the influence of specific complexes like the 26s proteasome in yeast where a small number of complexes dominate the k cores with large k and have a decisive effect on the strongest motifs with 6 8 nodes we also present zipf plots of counts versus rank they show broad distributions that are not power laws in contrast to the case when disconnected subgraphs are included

on the monotonicity of topological indices and the connectivity of a graph let i g be a topological index of a graph if i g e i g or i g e i g respectively for each edge e g then i g decreases or increases respectively with addition of edges in this paper we determine the extremal values of some monotonic topological indices in terms of the number of cut vertices or the number of cut edges or the vertex connectivity or the edge connectivity of a graph and characterize the corresponding extremal graphs among all graphs of order n

numerical computation and continuation of invariant manifolds connecting fixed points a numerical method for the computation of an invariant manifold that connects two fixed points of a vector field in mathbb r n is given extending the results of an earlier paper comput appl math 26 1989 pp 159 170 by the authors basically a boundary value problem on the real line is truncated to a finite interval the method applies in particular to the computation of heteroclinic orbits the emphasis is on the systematic computation of such orbits by continuation using the fact that the linearized operator of our problem is fredholm in appropriate banach spaces the general theory of approximation of nonlinear problems is employed to show that the errors in the approximate solution decay exponentially with the length of the approximating interval several applications are considered including the computation of traveling wave solutions to reaction diffusion problems computations were done using the software package auto

topological evolution in dense granular materials a complex networks perspective abstract one of the great challenges in the science of complex materials materials capable of emergent behavior such as self organized pattern formation is deciphering their inherent structural design principles as they deform in response to external loads we have been exploring the efficacy of techniques from complex networks to the study of dense granular materials as a means to i uncover such design principles and ii identify suitable metrics that quantify the evolution of structure during deformation herein we characterize the developing network structure and loss of connectivity in a quasistatically deforming granular medium from the perspective of complex networks attention is paid to the evolution of the contact and contact force networks at the local or mesoscopic level i e a particle and its immediate neighbors as well as the macroscopic level we explore network motifs and other topological properties at these multiple length scales in an attempt to find that which best correlates with the constitutive properties of nonaffine deformation and dissipation spatially and with respect to strain key processes or rearrangement events that cause loss of connectivity within the material domain e g microbanding and force chain buckling are investigated network statistics of these processes previously shown to be major sources of energy dissipation and nonaffine deformation are then tied to corresponding trends observed in the evolving macroscopic network it is shown that consideration of the unweighted contact network alone is insufficient to tie dissipation to loss of material connectivity

hilbert ℂ modules structural properties and applications to variational problems abstract we develop a theory of hilbert c over tilde modules which forms the core of a new functional analytic approach to algebras of generalized functions particular attention is given to nitely generated submodules projection operators representation theorems for c over tilde linear functionals and c over tilde sesquilinear forms we establish a generalized lax milgram theorem and use it to prove existence and uniqueness theorems for variational problems involving a generalized bilinear or sesquilinear form

random walks along the streets and canals in compact cities spectral analysis dynamical modularity information and statistical mechanics different models of random walks on the dual graphs of compact urban structures are considered analysis of access times between streets helps to detect the city modularity the statistical mechanics approach to the ensembles of lazy random walkers is developed the complexity of city modularity can be measured by an informationlike parameter which plays the role of an individual fingerprint of genius loci global structural properties of a city can be characterized by the thermodynamic parameters calculated in the random walk problem

extremal graphs for estrada indices abstract let g be a simple undirected connected graph the signless laplacian estrada laplacian estrada and estrada indices of a graph g is the sum of the exponentials of the signless laplacian eigenvalues laplacian eigenvalues and eigenvalues of g respectively the present work derives an upper bound for the estrada index of a graph as a function of its chromatic number in the family of graphs whose color classes have order not less than a fixed positive integer the graphs for which the upper bound is tight is obtained additionally an upper bound for the estrada index of the complement of a graph in the previous family of graphs with two color classes is given a nordhaus gaddum type inequality for the laplacian estrada index when g is a bipartite graph with color classes of order not less than 2 is presented moreover a sharp upper bound for the estrada index of the line graph and for the signless laplacian index of a graph in terms of connectivity is obtained

dependence centrality similarity measuring the diversity of profession levels of interests abstract to understand the relations between developers and software we study a collaborative coding platform from the perspective of networks including follower networks dependence networks and developer project bipartite networks through the analyzing of degree distribution pagerank and degree dependent nearest neighbors centrality we find that the degree distributions of all networks have a power law form except the out degree distributions of dependence networks the nearest neighbors centrality is negatively correlated with degree for developers but fluctuates around the average for projects in order to measure the diversity of profession levels of interests a new index called dependence centrality similarity is proposed and the correlation between dependence centrality similarity and degree is investigated the result shows an obvious negative correlations between dependence centrality similarity and degree

numerical continuation methods for studying periodic travelling wave wavetrain solutions of partial differential equations abstract periodic travelling waves wavetrains are an important solution type for many partial differential equations in this paper i review the use of numerical continuation for studying these solutions i discuss the calculation of the form and stability of a given periodic travelling wave and the calculation of boundaries in a two dimensional parameter plane for wave existence and stability i also describe the automated implementation of these numerical continuation procedures via the software package wavetrain http www ma hw ac uk wavetrain i conclude by discussing ongoing work on numerical continuation methods for determining the absolute stability of periodic travelling waves

on extremal bipartite unicyclic graphs abstract let u n be the set of connected bipartite unicyclic graphs with n vertices here we consider the extremal graphs in u n with respect to both the estrada index of themselves and the kirchhoff index of their complements we find that the first two and the last one unicyclic graphs in u n according to these two orderings are coincident

returnability in complex directed networks digraphs the concept of returnability is proposed for complex directed networks digraphs it can be seen as a generalization of the concept of reciprocity two measures of the returnability are introduced we establish closed formulas for the calculation of the returnability measures which are also related to the digraph spectrum the two measures are calculated for simple examples of digraphs as well as for real world complex directed networks and are compared with the reciprocity

graphs with the maximal estrada indices abstract two new transformations are proposed to compare the estrada indices between two graphs let n m be the set of the n m graphs where n and m are the numbers of vertices and edges respectively the graphs with the maximal estrada indices in n m are deduced by the new method for three cases namely unicyclic and bipartite unicyclic graphs m n bicyclic graphs m n 1 and the n m graphs without even cycles n 1 m 3 n 1 2

on extremal bipartite bicyclic graphs abstract let b n be the set of all connected bipartite bicyclic graphs with n vertices the estrada index of a graph g is defined as ee g i 1 n e i where 1 2 n are the eigenvalues of the adjacency matrix of g and the kirchhoff index of a graph g is defined as kf g i j r i j where r i j is the resistance distance between vertices v i and v j in g the complement of g is denoted by g in this paper sharp upper bound on ee g resp kf g of graph g in b n is established the corresponding extremal graphs are determined respectively furthermore by means of some newly created inequalities the graph g in b n with the second maximal ee g resp kf g is identified as well it is interesting to see that the first two bicyclic graphs in b n according to these two orderings are mainly coincident

complex networks and economics abstract as the first step toward a study of complex networks in economics we consider business networks categorized into bipartite networks through empirical study we show the possibility that business networks will fall into the scale free category to have an idea about the clustering coefficient and the averaged path length for bipartite networks we perform a one mode reduction we calculate the quantities for networks constructed by banks and companies before after a merger among banks and quantitatively show that the merger among banks increases the cliquishness of companies and decreases the typical separation between two companies

characterization of distributions having a value at a point in the sense of robinson abstract we characterize schwartz distributions having a value at a single point in the sense introduced by means of nonstandard analysis by a robinson they appear to be continuous functions in a neighborhood of the point this characterization improves a result by p loeb which assumes the everywhere existence of point values

on maximum estrada indices of bipartite graphs with some given parameters abstract the estrada index of a graph g is defined as e e g i 1 n e i where 1 2 n are the eigenvalues of the adjacency matrix of g in this paper we characterize the unique bipartite graph with maximum estrada index among bipartite graphs with given matching number and given vertex connectivity edge connectivity respectively

spectral centrality measures in complex networks complex networks are characterized by heterogeneous distributions of the degree of nodes which produce a large diversification of the roles of the nodes within the network several centrality measures have been introduced to rank nodes based on their topological importance within a graph here we review and compare centrality measures based on spectral properties of graph matrices we shall focus on pagerank pr eigenvector centrality ev and the hub and authority scores of the hits algorithm we derive simple relations between the measures and the in degree of the nodes in some limits we also compare the rankings obtained with different centrality measures

the increase in the resolvent energy of a graph due to the addition of a new edge abstract the resolvent energy er g of a graph g on n vertices whose adjacency matrix has eigenvalues 1 n is the sum of the reciprocals of the numbers n 1 n n we introduce the resolvent energy matrix r g and present an algorithm that produces this matrix this algorithm may also be used to update r g when new edges are introduced to g using the resolvent energy matrix r g we determine the increase in the resolvent energy er g of g caused by such edge additions made to g moreover we express this increase in terms of the characteristic polynomial of g and the characteristic polynomials of three vertex deleted subgraphs of g

strongly internal sets and generalized smooth functions abstract based on a refinement of the notion of internal sets in colombeau s theory so called strongly internal sets we introduce the space of generalized smooth functions a maximal extension of colombeau generalized functions generalized smooth functions as morphisms between sets of generalized points form a sub category of the category of topological spaces in particular they can be composed unrestrictedly

wave train selection behind invasion fronts in reaction diffusion predator prey models abstract wave trains or periodic travelling waves can evolve behind invasion fronts in oscillatory reaction diffusion models for predator prey systems although there is a one parameter family of possible wave train solutions in a particular predator invasion a single member of this family is selected sherratt 1998 13 has predicted this wave train selection using a system that is a valid approximation near a supercritical hopf bifurcation in the corresponding kinetics and when the predator and prey diffusion coefficients are nearly equal away from a hopf bifurcation or if the diffusion coefficients differ somewhat these predictions lose accuracy we develop a more general wave train selection prediction for a two component reaction diffusion predator prey system that depends on linearizations at the unstable homogeneous steady states involved in the invasion front this prediction retains accuracy farther away from a hopf bifurcation and can also be applied when the predator and prey diffusion coefficients are unequal we illustrate the selection prediction with its application to three models of predator invasions

applications of estrada indices and energy to a family of compound graphs abstract to track the gradual change of the adjacency matrix of a simple graph g into the signless laplacian matrix v nikiforov in 35 suggested the study of the convex linear combination a adjacency matrix a g d g 1 a g for 0 1 where a g and d g are the adjacency and the diagonal vertex degrees matrices of g respectively taking this definition as an idea the next matrix was considered for a b r the matrix a a b defined by a a b g a d g b a g extends the previous adjacency matrix this matrix is designated the a b adjacency matrix of g both adjacency matrices are examples of universal matrices already studied by w haemers in this paper we study the a b adjacency spectra for a family of compound graphs formed by disjoint balanced trees whose roots are identified to the vertices of a given graph in consequence new families of cospectral adjacency laplacian and signless laplacian graphs new hypoenergetic graphs graphs whose energy is less than its vertex number and new explicit formulae for estrada signless laplacian estrada and laplacian estrada indices of graphs were obtained moreover sharp upper bounds of the above indices for caterpillars in terms of length of the path and of the maximum number of its pendant vertices are given

invading wave fronts and their oscillatory wakes are linked by a modulated travelling phase resetting wave abstract periodic wave trains are the generic solution from for oscillatory reaction diffusion equations in one space dimension it has been shown previously that invasive wavefronts generate behind them a wave train with a different speed from that of the invasion sherratt physica d 70 1994 370 382 in this paper the mechanism of wave train generation is studied in detail for systems of two reaction diffusion equations close to a supercritical hopf bifurcation in the kinetics with equal diffusion coefficients a combination of analytical and numerical evidence is presented suggesting that the invasive front and wave train are separated by a modulated travelling wave of phase gradient in which phase singularities occur periodically this calculation leads to a prediction of the amplitude and speed of the wave train generated by invasion

approximating spectral sums of large scale matrices using stochastic chebyshev approximations computation of the trace of a matrix function plays an important role in many scientific computing applications including applications in machine learning computational physics e g lattice qua

spectral properties of unimodular lattice triangulations random unimodular lattice triangulations have been recently used as an embedded random graph model which exhibit a crossover behavior between an ordered large world and a disordered small world behavior using the ergodic pachner flips that transform such triangulations into another and an energy functional that corresponds to the degree distribution variance markov chain monte carlo simulations can be applied to study these graphs here we consider the spectra of the adjacency and the laplacian matrix as well as the algebraic connectivity and the spectral radius power law dependencies on the system size can clearly be identified and compared to analytical solutions for periodic ground states for random triangulations we find a qualitative agreement of the spectral properties with well known random graph models in the microcanonical ensemble analytical approximations agree with numerical simulations in the canonical ensemble a crossover behavior can be found for the algebraic connectivity and the spectral radius thus combining large world and small world behavior in one model the considered spectral properties can be applied to transport problems on triangulation graphs and the crossover behavior allows a tuning of important transport quantities

algebraic theory of colombeauʼs generalized numbers abstract let k denote the commutative ring of colombeau s full generalized numbers endowed with scarpalezos sharp topology it becomes a topological ring we study the algebraic and topological properties of this topological ring in particular we prove that the group of units of k is dense in the sharp topology determine its boolean algebra show that it has minimal primes describe them completely which results in a complete classification of the maximal ideals from the description of the prime and maximal ideals it becomes clear that they should be determined by certain ultra filters

stability of synchronized states in networks of phase oscillators the stability of synchronized states frequency locked states in networks of phase oscillators is investigated for several network topologies it is shown that for some topologies there is more than one stable synchronized state according to the sign of coupling strength between oscillators it is also shown that in some cases the synchronized state corresponds to zero order parameter

on the r uniform linear hypertrees with extremal estrada indices abstract let h be a hypergraph with n vertices the estrada index of h is defined as the sum of e 1 e n where 1 n are the eigenvalues of the adjacency matrix of h let t n r be the set of r uniform linear hypertrees with n vertices and t n r the subset of t n r in which each hypergraph has a largest vertex degree where r 3 and 2 n 1 r 1 several new transformations for studying the estrada indices of hypergraphs are introduced by these transformations the hypertree with the smallest estrada index is determined in t n r and the hypertrees with the smallest the second smallest the largest and the second largest estrada indices are derived among t n r

computing absolute and essential spectra using continuation a continuation approach to the computation of essential and absolute spectra of dierential operators on the real line is presented the advantage of this approach compared with direct eigenvalue computations for the discretized operator are the ecient and accurate computation of selected parts of the spectrum typically those near the imaginary axis and the option to compute nonlinear travelling waves and selected eigenvalues or other stability indicators simultaneously in order to locate accurately the onset to instability we also discuss the implementation and usage of this approach with the software package auto and provide example computations for the fitzhugh nagumo and the complex ginzburg landau equation

locally convex topological algebras of generalized functions compactness and nuclearity in a nonlinear context in this paper we introduce hausdorff locally convex algebra topologies on subalgebras of the whole algebra of nonlinear generalized functions these topologies are strong duals of frechet schwartz space topologies and even strong duals of nuclear frechet space topologies in particular any bounded set is relatively compact and one benefits from all deep properties of nuclearity these algebras of generalized functions contain most of the classical irregular functions and distributions they are obtained by replacing the mathematical tool of c functions in the original version of nonlinear generalized functions by the far more evolved tool of holomorphic functions this paper continues the nonlinear theory of generalized functions in which such locally convex topological properties were strongly lacking up to now

on maximum estrada indices of graphs with given parameters abstract for a graph g with eigenvalues 1 2 n its estrada index is defined as ee g i 1 n e i we determine the unique graphs with maximum estrada indices among graphs with given number of cut vertices connectivity and edge connectivity respectively

topological structural classes of complex networks we use theoretical principles to study how complex networks are topologically organized at large scale using spectral graph theory we predict the existence of four different topological structural classes of networks these classes correspond respectively to highly homogenous networks lacking structural bottlenecks networks organized into highly interconnected modules with low inter community connectivity networks with a highly connected central core surrounded by a sparser periphery and networks displaying a combination of highly connected groups quasicliques and groups of nodes partitioned into disjoint subsets quasibipartites here we show by means of the spectral scaling method that these classes really exist in real world ecological biological informational technological and social networks we show that neither of three network growth mechanisms random with uniform distribution preferential attachment and random with the same degree sequence as real network is able to reproduce the four structural classes of complex networks these models reproduce two of the network classes as a function of the average degree but completely fail in reproducing the other two classes of networks

random neighborhood graphs as models of fracture networks on rocks structural and dynamical analysis abstract we propose a new model to account for the main structural characteristics of rock fracture networks rfns the model is based on a generalization of the random neighborhood graphs to consider fractures embedded into rectangular spaces we study a series of 29 real world rfns and find the best fit with the random rectangular neighborhood graphs rrngs proposed here we show that this model captures most of the structural characteristics of the rfns and allows a distinction between small and more spherical rocks and large and more elongated ones we use a diffusion equation on the graphs in order to model diffusive processes taking place through the channels of the rfns we find a small set of structural parameters that highly correlates with the average diffusion time in the rfns we found analytically some bounds for the diameter and the algebraic connectivity of these graphs that allow to bound the diffusion time in these networks we also show that the rrngs can be used as a suitable model to replace the rfns in the study of diffusion like processes indeed the diffusion time in rfns can be predicted by using structural and dynamical parameters of the rrngs finally we also explore some potential extensions of our model to include variable fracture apertures the possibility of long range hops of the diffusive particles as a way to account for heterogeneities in the medium and possible superdiffusive processes and the extension of the model to 3 dimensional space

the estrada index of evolving graphs abstract the estrada index of a simple graph g with adjacency matrix a is defined as ee g trace e a it can be used in a range of fields as an efficient measuring tool an evolving graph g 1 g 2 g n is a time ordered sequence of graphs over a fixed set of vertices by using the natural definition of a walk on the evolving graph that respects the arrow of time we extend the static estrada index to accommodate this promising dynamic setting we find that although asymmetry is raised intrinsically by time s arrow the dynamic estrada index is order invariant we establish some lower and upper bounds for the dynamic estrada index in terms of the numbers of vertices and edges illustrative examples are worked out to demonstrate the computations involved

estimating the estrada index abstract let g be a graph on n vertices and let 1 2 n be its eigenvalues the estrada index of g is a recently introduced graph invariant defined as ee i 1 n e i we establish lower and upper bounds for ee in terms of the number of vertices and number of edges also some inequalities between ee and the energy of g are obtained

the effects of unequal diffusion coefficients on periodic travelling waves in oscillatory reaction diffusion systems abstract many oscillatory biological systems show periodic travelling waves these are often modelled using coupled reaction diffusion equations however the effects of different movement rates diffusion coefficients of the interacting components on the predictions of these equations are largely unknown here we investigate the ways in which varying the diffusion coefficients in such equations alters the wave speed time period wavelength amplitude and stability of periodic wave solutions we focus on two sets of kinetics that are commonly used in ecological applications lambda omega equations which are the normal form of an oscillatory coupled reaction diffusion system close to a supercritical hopf bifurcation and a standard predator prey model our results show that changing the ratio of the diffusion coefficients can significantly alter the shape of the one parameter family of periodic travelling wave solutions the position of the boundary between stable and unstable waves also depends on the ratio of the diffusion coefficients in all cases stability changes through an eckhaus sideband instability these effects are always symmetrical in the two diffusion coefficients for the lambda omega equations but are asymmetric in the predator prey equations especially when the limit cycle of the kinetics is of large amplitude in particular there are two separate regions of stable waves in the travelling wave family for some parameter values in the predator prey scenario our results also show the existence of a one parameter family of travelling waves but not necessarily a hopf bifurcation for all values of the diffusion coefficients simulations of the full partial differential equations reveals that varying the ratio of the diffusion coefficients can significantly change the properties of periodic travelling waves that arise from particular wave generation mechanisms and our analysis of the travelling wave families assists in the understanding of these effects

on estrada index of trees abstract let g be a graph on n vertices and let 1 2 n be its eigenvalues the estrada index is defined as ee g i 1 n e i we determine the unique tree with maximum estrada index among the trees on n vertices with given matching number and the unique tree with maximum estrada index among the trees on n vertices with fixed diameter for n 1 3 we also determine the tree with maximum estrada index among the trees on n vertices with maximum degree it gives a partial solution to the conjecture proposed by ilic and stevanovic in ref 14

the estrada index of trees the estrada index of a graph g is defined as ee g i 1ne i where 1 2 n are the eigenvalues of its adjacency matrix we determine the unique tree with maximum estrada index among the set of trees with given number of pendant vertices as applications we determine trees with maximum estrada index among the set of trees with given matching number independence number and domination number respectively finally we give a proof of a conjecture in j li x li l wang the minimal estrada index of trees with two maximum degree vertices match commun math comput chem 64 2010 799 810 on trees with minimum estrada index among the set of trees with two adjacent vertices of maximum degree

bounds of the extended estrada index of graphs abstract let g be a graph on n vertices and 1 2 n the eigenvalues of its extended adjacency matrix the extended estrada index ee ex is defined as the sum of the terms e i i 1 2 n in this paper we establish lower and upper bounds for ee ex in terms of the number of vertices and the number of edges and characterize the extremal graphs also the bounds for ee ex of some special graphs are obtained

the estrada index of unicyclic graphs abstract the estrada index of a graph g is defined as ee g i 1 n e i where 1 2 n are the eigenvalues of g we determine the unique unicyclic graph with maximum estrada index the unique bipartite unicyclic graph with maximum estrada index and the unique bipartite unicyclic graph of a given bipartition with maximum estrada index we also show that only two graphs are possible to be the unicyclic graph s with minimum estrada index