novel inequalities for generalized graph entropies graph energies and topological indices the entropy of a graph is an information theoretic quantity for measuring the complexity of a graph after shannon introduced the entropy to information and communication many generalizations of the entropy measure have been proposed such as renyi entropy and daroczy entropy in this article we prove accurate connections inequalities between generalized graph entropies graph energies and topological indices additionally we obtain some extremal properties of nine generalized graph entropies by employing graph energies and topological indices

brief paper design of highly synchronizable and robust networks in this paper the design of highly synchronizable sparse and robust dynamical networks is addressed better synchronizability means faster synchronization of the oscillators sparsity means a low ratio of links per nodes and robustness refers to the resilience of a network to the random failures or intentional removal of some of the nodes links golden spectral dynamical networks graphs are those for which the spectral spread the difference between the largest and smallest eigenvalues of the adjacency matrix is equal to the spectral gap the difference between the two largest eigenvalues of the adjacency matrix multiplied by the square of the golden ratio these networks display the property of small worldness are very homogeneous and have large isoperimetric expansion constant together with a very high synchronizability and robustness to failures of individual oscillators in particular the regular bipartite dynamical networks reported here by the first time have the best possible expansion and consequently are the most robust ones against node link failures or intentional attacks

synchronizable nodes in networked systems synchronization appears in networked systems as a phenomenon of interacting nodes the notion of synchronizability describes the property that the nodes can be forced to a synchronous behaviour by an appropriate interconnection structure whereas in earlier papers synchronizability has been investigated in terms of both the properties of the nodes and of the interconnection topology this paper shows that for linear coupled systems synchronizability is a property of the node dynamics alone if the nodes do not satisfy the necessary and sufficient synchronizability condition derived no interaction structure can ensure a synchronous behaviour of the networked system on the other hand if this condition is satisfied several interconnection structures given in this paper ensure synchronism globally in the whole state space

synchronizability of complex networks recent interest in the study of networks associated with complex systems has led to a better understanding of the factors and parameters relating the topology of the associated networks with the dynamics of the systems and in particular their synchronization in this paper by using known and new results from spectral graph theory we characterize relevant factors which affect the synchronization of complex networks

on the laplacian energy like invariant abstract let g be a connected graph of order n with laplacian eigenvalues 1 2 n 1 n 0 the laplacian energy like invariant of the graph g is defined as lel lel g i 1 n 1 i lower and upper bounds for lel are obtained in terms of n number of edges maximum vertex degree and number of spanning trees

on the tree with diameter 4 and maximal energy the energy of a simple graph is defined as the sum of the absolute values of all the eigenvalues of its adjacency matrix jianping found the structure of trees which have maximal energy among all trees of diameter 4 whose centers are of degree t this paper generalizes the result we determine the structure of the general tree of diameter 4 and maximal energy

network synchronization diffusion and the paradox of heterogeneity many complex networks display strong heterogeneity in the degree connectivity distribution heterogeneity in the degree distribution often reduces the average distance between nodes but paradoxically may suppress synchronization in networks of oscillators coupled symmetrically with uniform coupling strength here we offer a solution to this apparent paradox our analysis is partially based on the identification of a diffusive process underlying the communication between oscillators and reveals a striking relation between this process and the condition for the linear stability of the synchronized states we show that for a given degree distribution the maximum synchronizability is achieved when the network of couplings is weighted and directed and the overall cost involved in the couplings is minimum this enhanced synchronizability is solely determined by the mean degree and does not depend on the degree distribution and system size numerical verification of the main results is provided for representative classes of small world and scale free networks

a spectral approach to the randic index abstract the higher randic index rt of a simple graph is defined as r t v i 1 v i 2 v i t 1 1 i 1 i 2 i t 1 where i denotes the degree of the vertex vi and v i 1 v i 2 v i t 1 runs over all paths of length t in in this paper we introduce a suitable version of the laplacian of a graph and we formulated r1 in terms of its spectrum moreover bounds on r2 from the eigenvalues either the adjacency matrix or the laplacian matrix of the graph are obtained in the paper

spectral properties of geometric arithmetic index the concept of geometric arithmetic index was introduced in the chemical graph theory recently but it has shown to be useful one of the main aims of algebraic graph theory is to determine how or whether properties of graphs are reflected in the algebraic properties of some matrices the aim of this paper is to study the geometric arithmetic index ga1 from an algebraic viewpoint since this index is related to the degree of the vertices of the graph our main tool will be an appropriate matrix that is a modification of the classical adjacency matrix involving the degrees of the vertices moreover using this matrix we define a ga laplacian matrix which determines the geometric arithmetic index of a graph and satisfies properties similar to the ones of the classical laplacian matrix

clustering and the synchronization of oscillator networks by manipulating the clustering coefficient of a network without changing its degree distribution we examine the effect of clustering on the synchronization of phase oscillators on networks with poisson and scale free degree distributions for both types of networks increased clustering hinders global synchronization as the network splits into dynamical clusters that oscillate at different frequencies surprisingly in scale free networks clustering promotes the synchronization of the most connected nodes hubs even though it inhibits global synchronization as a result they show an additional advanced transition instead of a single synchronization threshold this cluster enhanced synchronization of hubs may be relevant to the brain that is scale free and highly clustered

an invariance principle for maps with polynomial decay of correlations we give a general method of deriving statistical limit theorems such as the central limit theorem and its functional version in the setting of ergodic measure preserving transformations this method is applicable in situations where the iterates of discrete time maps display a polynomial decay of correlations

subexponential decay of correlations we describe a method for proving subexponential lower bounds for correlations functions and apply it to study decay of correlations for maps with countable markov partitions one result is that ls young s upper estimates y2 are optimal in many situations our method is based on a general result concerning the asymptotics of renewal sequences of bounded operators acting on banach spaces which we apply to the iterates of the transfer operator

stabilizing weighted complex networks real networks often consist of local units which interact with each other via asymmetric and heterogeneous connections in this paper the v stability problem is investigated for a class of asymmetric weighted coupled networks with nonidentical node dynamics which includes the unweighted network as a special case pinning control is suggested to stabilize such a coupled network the complicated stabilization problem is reduced to measuring the semi negative property of the characteristic matrix which embodies not only the network topology but also the node self dynamics and the control gains it is found that network stabilizability depends critically on the second largest eigenvalue of the characteristic matrix the smaller the second largest eigenvalue is the more the network is pinning controllable numerical simulations of two representative networks composed of non chaotic systems and chaotic systems respectively are shown for illustration and verification

structural and spectral properties of a family of deterministic recursive trees rigorous solutions as one of the most significant models the uniform recursive tree urt has found many applications in a variety of fields in this paper we study rigorously the structural features and spectral properties of the adjacency matrix for a family of deterministic uniform recursive trees durts that are deterministic versions of urt firstly from the perspective of complex networks we investigate analytically the main structural characteristics of durts and obtain the accurate solutions for these properties which include degree distribution average path length distribution of node betweenness and degree correlations then we determine the complete eigenvalues and their corresponding eigenvectors of the adjacency matrix for durts our research may shed light on a better understanding of the features for urt also the analytical methods used here are capable of being extended to many other deterministic networks making the precise computation of their properties especially the full spectrum characteristics possible

factors that predict better synchronizability on complex networks while shorter characteristic path length has in general been believed to enhance synchronizability of a coupled oscillator system on a complex network the suppressing tendency of the heterogeneity of the degree distribution even for shorter characteristic path length has also been reported to see this we investigate the effects of various factors such as the degree characteristic path length heterogeneity and betweenness centrality on synchronization and find a consistent trend between the synchronization and the betweenness centrality the betweenness centrality is thus proposed as a good indicator for synchronizability

markov approximations and decay of correlations for anosov flows we develop markov approximations for very general suspension flows based on this we obtain a stretched exponential bound on time correlation functions for 3 d anosov flows that verify uniform nonintegrability of foliations these include contact anosov flows and geodesic flows on compact surfaces of variable negative curvature our bound on correlations is stable under small smooth perturbations

relations between average distance heterogeneity and network synchronizability abstract by using the random interchanging algorithm we investigate the relations between average distance standard deviation of degree distribution and synchronizability of complex networks we find that both increasing the average distance and magnifying the degree deviation will make the network synchronize harder only the combination of short average distance and small standard deviation of degree distribution ensures strong synchronizability some previous studies assert that the maximal betweenness is the right quantity to estimate network synchronizability the larger the maximal betweenness the poorer the network synchronizability here we address an interesting case which strongly suggests that the single quantity maximal betweenness may not give a comprehensive description of network synchronizability

convergence to equilibrium for intermittent symplectic maps we investigate a class of area preserving non uniformly hyperbolic maps of the two torus first we establish some results on the regularity of the invariant foliations then we use this knowledge to estimate the rate of mixing

spectral reconstruction of complex networks abstract in this paper we study the reconstruction of a network topology from the eigenvalues of its laplacian matrix we introduce a simple cost function and consider the tabu search combinatorial optimization method while comparing its performance when reconstructing different categories of networks random regular small world scale free and clustered from their eigenvalues we show that this combinatorial optimization method together with the information contained in the laplacian spectrum allows an exact reconstruction of small networks and leads to good approximations in the case of networks with larger orders we also show that the method can be used to generate a quasi optimal topology for a network associated to a dynamic process like in the case of metabolic or protein protein interaction networks of organisms

spectral bounds for the betweenness of a graph in this paper we find spectral bounds laplacian matrix for the vertex and the edge betweenness of a graph we also relate the edge betweenness with the isoperimetric number and the edge forwarding and edge expansion indices of the graph allowing a new upper bound on its diameter the results are of interest as they can be used in the study of communication properties of real networks in particular for dynamical processes taking place on them broadcasting network synchronization virus spreading etc

synchronization of complex dynamical networks with switching topology a switched system point of view in this paper we study the synchronization problem for complex dynamical networks with switching topology from a switched system point of view the synchronization problem is transformed into the stability problem for time varying switched systems we address two basic problems synchronization under arbitrary switching topology and synchronization via design of switching within a pre given collection of topologies when synchronization cannot be achieved by using any topology alone in this collection for the both problems we first establish synchronization criteria for general connection topology then under the condition of simultaneous triangularization of the connection matrices a common lyapunov function for the first problem and a single lyapunov and multiple lyapunov functions for the second problem are systematically constructed respectively by those of several lower dimensional dynamic systems in order to achieve synchronization using multiple lyapunov functions a stability condition and switching law design method for time varying switched systems are also presented which avoid the usual non increasing condition

note on two generalizations of the randic index for a given graph g the well known randic index of g introduced by milan randic in 1975 is defined as r g u v e g d u d v 1 2 where the sum is taken over all edges uv and du denotes the degrees of u bollobas and erdos generalized this index by replacing 1 2 with any real number which is called the general randic index dvo ak et al introduced a modified version of randic index r g u v e g max d u d v 1 based on this recently knor et al introduced two generalizations r g u v e g min d u d v and r g u v e g max d u d v for any real number clearly the former is a lower bound for the general randic index and the latter is its upper bound knor et al studied extremal values of r g and r g and concluded some open problems in this paper we consider the open problems and give some comments and results some results for chemical trees are obtained