cored hypergraphs power hypergraphs and their laplacian h eigenvalues abstract in this paper we introduce the class of cored hypergraphs and power hypergraphs and investigate the properties of their laplacian h eigenvalues from an ordinary graph one may generate a k uniform hypergraph called the kth power hypergraph of that graph power hypergraphs are cored hypergraphs but not vice versa sunflowers loose paths and loose cycles are power hypergraphs while squids generalized loose s paths and loose s cycles for 2 s k 2 are cored hypergraphs but not power graphs in general we show that the largest laplacian h eigenvalue of an even uniform cored hypergraph is equal to its largest signless laplacian h eigenvalue especially we find out these largest h eigenvalues for even uniform squids moreover we show that the largest laplacian h eigenvalue of an odd uniform squid loose path and loose cycle is equal to the maximum degree i e 2 we also compute the laplacian h spectra of the class of sunflowers when k is odd the laplacian h spectra of the loose cycle of size 3 and the loose path of length 3 are characterized as well

nonlinear oscillator with discontinuity by generalized harmonic balance method a generalized harmonic balance method is used to calculate the periodic solutions of a nonlinear oscillator with discontinuities for which the elastic force term is proportional to sgn x this method is a modification of the generalized harmonic balance method in which analytical approximate solutions have rational form this approach gives us not only a truly periodic solution but also the frequency of the motion as a function of the amplitude of oscillation we find that this method works very well for the whole range of amplitude of oscillation in the case of the antisymmetric piecewise constant force oscillator and excellent agreement of the approximate frequencies with the exact one has been demonstrated and discussed for the second order approximation we have shown that the relative error in the analytical approximate frequency is 0 24 we also compared the fourier series expansions of the analytical approximate solution and the exact one comparison of the result obtained using this method with the exact ones reveals that this modified method is very effective and convenient

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

static and dynamic stability results for a class of three dimensional configurations of kirchhoff elastic rods we analyze the dynamical stability of a naturally straight inextensible and unshearable elastic rod under tension and controlled end rotation within the kirchhoff model in three dimensions the cases of clamped boundary conditions and isoperimetric constraints are treated separately we obtain explicit criteria for the static stability of arbitrary extrema of a general quadratic strain energy we exploit the equivalence between the total energy and a suitably defined norm to prove that local minimizers of the strain energy under explicit hypotheses are stable in the dynamic sense due to liapounov we also extend our analysis to damped systems to show that static equilibria are dynamically stable in the liapounov sense in the presence of a suitably defined local drag force

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

towards a classification of euler kirchhoff filaments euler kirchhoff filaments are solutions of the static kirchhoff equations for elastic rods with circular cross sections these equations are known to be formally equivalent to the euler equations for spinning tops this equivalence is used to provide a classification of the different shapes a filament can assume explicit formulas for the different possible configurations and specific results for interesting particular cases are given in particular conditions for which the filament has points of self intersection self tangency vanishing curvature or when it is closed or localized in space are provided the average properties of generic filaments are also studied they are shown to be equivalent to helical filaments on long length scales

harmonic balance approach to limit cycles for nonlinear jerk equations the method of harmonic balance hb is employed to estimate the attributes of limit cycles of some nonlinear third order jerk differential equations which are parity invariant but not time reversal invariant two examples with cubic nonlinearities show that the hb method can give good values for the frequency and both the velocity and displacement amplitudes of a period one limit cycle

solution of a duffing harmonic oscillator by the method of harmonic balance abstract the first order harmonic balance method via first fourier coefficient is used to construct an approximate frequency amplitude relation for a duffing harmonic oscillator this relation is in agreement with the result obtained by the ritz procedure

topological generalizations of network motifs biological and technological networks contain patterns termed network motifs which occur far more often than in randomized networks network motifs were suggested to be elementary building blocks that carry out key functions in the network it is of interest to understand how network motifs combine to form larger structures to address this we present a systematic approach to define motif generalizations families of motifs of different sizes that share a common architectural theme to define motif generalizations we first define roles in a subgraph according to structural equivalence for example the feedforward loop triad a motif in transcription neuronal and some electronic networks has three roles an input node an output node and an internal node the roles are used to define possible generalizations of the motif the feedforward loop can have three simple generalizations based on replicating each of the three roles and their connections we present algorithms for efficiently detecting motif generalizations we find that the transcription networks of bacteria and yeast display only one of the three generalizations the multi output feedforward generalization in contrast the neuronal network of c elegans mainly displays the multi input generalization forward logic electronic circuits display a multi input multi output hybrid thus networks which share a common motif can have very different generalizations of that motif using mathematical modeling we describe the information processing functions of the different motif generalizations in transcription neuronal and electronic networks

subgraphs and network motifs in geometric networks many real world networks describe systems in which interactions decay with the distance between nodes examples include systems constrained in real space such as transportation and communication networks as well as systems constrained in abstract spaces such as multivariate biological or economic data sets and models of social networks these networks often display network motifs subgraphs that recur in the network much more often than in randomized networks to understand the origin of the network motifs in these networks it is important to study the subgraphs and network motifs that arise solely from geometric constraints to address this we analyze geometric network models in which nodes are arranged on a lattice and edges are formed with a probability that decays with the distance between nodes we present analytical solutions for the numbers of all three and four node subgraphs in both directed and nondirected geometric networks we also analyze geometric networks with arbitrary degree sequences and models with a bias for directed edges in one direction scaling rules for scaling of subgraph numbers with system size lattice dimension and interaction range are given several invariant measures are found such as the ratio of feedback and feed forward loops which do not depend on system size dimension or connectivity function we find that network motifs in many real world networks including social networks and neuronal networks are not captured solely by these geometric models this is in line with recent evidence that biological network motifs were selected as basic circuit elements with defined information processing functions

nonlinear dynamics of filaments ii nonlinear analysis abstract the linear stability analysis of the full time dependent kirchhoff equations for elastic filaments gives precise information about possible dynamical instabilities the associated dispersion relations derived in the preceding paper provides the selection mechanism for the shapes selected by highly unstable filaments here we perform a nonlinear analysis and derive new amplitude equations which describe the dynamics above the instability threshold the straight filament is studied in detail and the motion is shown to be described by a pair of nonlinear klein gordon equations which couple the local deformation amplitude to the twist density of particular interest is the effect of boundary conditions on the instability threshold it is shown that with suitable choice of boundary conditions the threshold of instability is delayed we also show the existence of pulse like and front like traveling wave solutions

comments on a finite extensibility nonlinear oscillator abstract the aim of this comment is to provide more information about the study of the dynamics of a finite extensibility nonlinear oscillator conducted by febbo m febbo a finite extensibility nonlinear oscillator applied mathematics and computation 217 2011 6464 6475 we show that the linearized harmonic balance method is not sufficiently adequate for this oscillator and that the harmonic balance method hbm without linearization provides better results we also discuss what happens when the oscillation amplitude approaches 1 and why the harmonic balance method does not give optimum results for these values of the oscillation amplitude the periodic solution becomes markedly anharmonic and is almost straight between x a and x a with negative slope and between x a and x a with positive slope finally a heuristic solution is proposed which is adequate for the whole amplitude range 0 a a

application of the harmonic balance method to a nonlinear oscillator typified by a mass attached to a stretched wire abstract the first order harmonic balance method via the first fourier coefficient is used to construct two approximate frequency amplitude relations for a conservative nonlinear oscillatory system in which the restoring force has an irrational form this system corresponds to the motion of a mass attached to a stretched wire two procedures are used to approximately solve the nonlinear differential equation in the first the differential equation is rewritten in a form that does not contain the square root expression while in the second the differential equation is solved directly the approximate frequency obtained using the second procedure is more accurate than the frequency obtained with the first due to the fact that in the second procedure application of the harmonic balance method produces an infinite set of harmonics while in the first procedure only two harmonics are produced both approximate frequencies are valid for the complete range of oscillation amplitudes and excellent agreement of the approximate frequencies with the exact one are demonstrated and discussed the discrepancy between the second approximate frequency and the exact one never exceeds 2 2

stability of nonlinearly elastic rods this paper describes previously unknown stabilities and instabilities of planar equilibrium configurations of a nonlinearly elastic rod that is buckled under the action of a dead load the governing equations are derived from variational principles including ones of isoperimetric type properties of stability are accordingly determined by study of the second variation stabilities to deformations both in the plane and out of the plane are considered among the newly discovered properties are secondary bifurcation from the first buckled mode marked differences between stability to two dimensional and to three dimensional variations and the stabilizing influence of resistance to twist in the isoperimetric examples the analysis makes crucial use of a novel device to account for the dependence of the second variation on constraints

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 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

clustering of correlated networks we obtain the clustering coefficient the degree dependent local clustering and the mean clustering of networks with arbitrary correlations between the degrees of the nearest neighbor vertices the resulting formulas allow one to determine the nature of the clustering of a network

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

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

solution of a quadratic nonlinear oscillator by the method of harmonic balance abstract this paper deals with the quadratic nonlinear oscillator qno x x e x 2 0 this oscillator is compared with the nonlinear oscillators x x e x x 0 which are solved by the method of harmonic balance therefore we obtain the first approximate solution to the qno this solution is more accurate than the second approximate solution obtained by the lindstedt poincare method for qnos it has been pointed out that the angular frequency for x 0 should be different from the angular frequency for x 0 if solutions of higher degree of accuracy are required

mean field theory for clustering coefficients in barabasi albert networks we applied a mean field approach to study clustering coefficients in barabasi albert ba networks we found that the local clustering in ba networks depends on the node degree analytic results have been compared to extensive numerical simulations finding a very good agreement for nodes with low degrees clustering coefficient of a whole network calculated from our approach perfectly fits numerical data

on the determination of approximate periodic solutions of some non linear ode abstract we propose a numerical symbolic method for the approximation of periodic solutions of a type of non linear ode the efficiency of our method is contrasted with the harmonic balance method and with another one which combines the differential transformation method with pade approximants on a non trivial example the relativistic oscillator it is shown that our method is computationally more reliable

bounds on the largest eigenvalues of trees with a given size of matching very little is known about upper bound for the largest eigenvalue of a tree with a given size of matching in this paper we find some upper bounds for the largest eigenvalue of a tree in terms of the number of vertices and the size of matchings which improve some known results

higher order clustering coefficients in barabasi albert networks abstract higher order clustering coefficients c x are introduced for random networks the coefficients express probabilities that the shortest distance between any two nearest neighbours of a certain vertex i equals x when one neglects all paths crossing the node i using c x we found that in the barabasi albert ba model the average shortest path length in a node s neighbourhood is smaller than the equivalent quantity of the whole network and the remainder depends only on the network parameter m our results show that small values of the standard clustering coefficient in large ba networks are due to random character of the nearest neighbourhood of vertices in such networks

on the dynamics of rods in the theory of kirchhoff and clebsch we discuss here the dynamical equations of a theory of elastic rods that is due to kmcm ioff 1 2 and cl bsc 3 4 this properly invariant theory is applicable to motions in which the strains relative to an undistorted configuration remain small although rotations may be large it is constructed to be a first order theory i e a theory that is complete to within an error of order two in an appropriate dimensionless measure of thickness curvature twist and extension in a first order theory of thin rods one can treat the rod as inextensible and we do so here at the outset thus at each time t the arc length parameter s for the axial curve t is employed as a material coordinate i e a parameter whose value at a material point is constant in time and not only the resultant of the shearing forces on a cross section but also the tension in the rod are reactive quantities not given by constitutive equations consider for a moment a rod that is naturally prismatic and dynamically symmetric i e a rod that in an undistorted stress free configuration is a cylinder whose directrix although not necessarily a circle bounds a figure with equal principal moments of inertia a motion of the rod is said to be planar and twist flee i e a motion of pure flexure if it is such that t lies at all times in a fixed plane which contains a principal axis of inertia of each cross section for such a motion we employ a fixed cartesian coordinate system on and for consistency with a discussion of more general motions to be given later in this paper we call the abscissa z and the ordinate x we may write fz s t fx s t for the z and x components of the resultant force f at time t on the cross section with arc length coordinate s the motion of the rod may be described by giving the z x coordinates of the points on t as functions of s and t with o s t the counterclockwise angle from the z axis to the tangent of t at s we have

a finite extensibility nonlinear oscillator abstract the dynamics of a finite extensibility nonlinear oscillator feno is studied analytically by means of two different approaches a generalized decomposition method gdm and a linearized harmonic balance procedure lhb from both approaches analytical approximations to the frequency of oscillation and periodic solutions are obtained which are valid for a large range of amplitudes of oscillation within the generalized decomposition method two different versions are presented which provide different kinds of approximate analytical solutions in the first version it is shown that the truncation of the perturbation solution up to the third order provides a remarkable degree of accuracy for almost the whole range of amplitudes the second version which expands the nonlinear term in taylor s series around the equilibrium point exhibits a little lower degree of accuracy but it supplies an infinite series as the approximate solution on the other hand a linearized harmonic balance method is also employed and the comparison between the approximate period and the exact one numerically calculated is slightly better than that obtained by both versions of the gdm in general the agreement between the results obtained by the three methods and the exact solution numerically integrated for amplitudes a between 0 a 0 9 is very good both for the period and the amplitude of oscillation for the rest of the amplitude range 0 9 a l 2 error demonstrates that all three approximations do not represent a good description for the feno and higher order perturbation solutions are needed instead as a complement very accurate asymptotic representations of the period are provided for the whole range of amplitudes of oscillation

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

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

on the kirchhoff index of the complement of a bipartite graph abstract the kirchhoff index of a graph g is defined as kf g 1 2 i 1 n j 1 n r ij where r ij is the resistance distance between v i and v j let g be a simple bipartite graph g the complement graph of g first we get an expression for kf g in terms of the numbers of closed walks of subdivision graph of g then as an application we determine the trees with the first and second maximum and minimum values of kf t

investigation on non linear vibration in arched beam for bridges construction via agm method analyzing and modeling the vibrational behavior of arched bridges during earthquake in order to decrease the exerted damages to the structure is a very hard task to do in the present paper this item has been performed analytically for the first time due to the importance of arched bridges building as a great structure in the human being civilization and its specifications such as transferring vertical loads to its arcs and the lack of bending moments and shearing forces this case study is devoted to this special issue therefore the nonlinear vibration of arched bridges has been modeled and simulated by an arched beam with harmonic vertical loads and its behavior has been investigated by analyzing a nonlinear partial differential equation governing on the system it is notable that the procedure has been done analytically by agm akbari ganji s method it has found that angular frequency is increased significantly by escalating the amount of the center outlet furthermore comparisons have been made between the obtained results by numerical method rkf 45 and agm in order to assess the scientific validity obtained results indicate the precision of the achieved answers by agm method

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

isoperimetric conjugate points with application to the stability of dna minicircles a conjugate point test determining an index of the constrained second variation in one dimensional isoperimetric calculus of variations problems is described the test is then implemented numerically to determine stability properties of equilibria within a continuum mechanics model of dna minicircles

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

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

solution of the relativistic an harmonic oscillator using the harmonic balance method this work was supported by the ministerio de educacion y ciencia spain under project fis2005 05881 c02 02 and by the generalitat valenciana spain under project acomp 2007 020

the architecture of complex weighted networks networked structures arise in a wide array of different contexts such as technological and transportation infrastructures social phenomena and biological systems these highly interconnected systems have recently been the focus of a great deal of attention that has uncovered and characterized their topological complexity along with a complex topological structure real networks display a large heterogeneity in the capacity and intensity of the connections these features however have mainly not been considered in past studies where links are usually represented as binary states i e either present or absent here we study the scientific collaboration network and the world wide air transportation network which are representative examples of social and large infrastructure systems respectively in both cases it is possible to assign to each edge of the graph a weight proportional to the intensity or capacity of the connections among the various elements of the network we define appropriate metrics combining weighted and topological observables that enable us to characterize the complex statistical properties and heterogeneity of the actual strength of edges and vertices this information allows us to investigate the correlations among weighted quantities and the underlying topological structure of the network these results provide a better description of the hierarchies and organizational principles at the basis of the architecture of weighted networks

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

subgraph centrality and clustering in complex hyper networks abstract the representation of complex systems as networks is inappropriate for the study of certain problems we show several examples of social biological ecological and technological systems where the use of complex networks gives very limited information about the structure of the system consequently we extend the concepts of subgraph centrality and clustering for complex networks represented by hypergraphs complex hyper networks the first parameter characterizes the node participation in different sub hypergraphs and the second one characterizes the transitivity in the hyper network through the proportion of hyper triangles to paths of length two another measure characterizing the formation of triples of mutually adjacent groups in the hyper network is also introduced all of these characteristics are studied in three different hyper networks a scientific collaboration hyper network an ecological competition hyper network and the hyper network formed by the american corporate elite in 1999

nonlinear dynamical theory of the elastica the dynamical behaviour of a slender rod is analyzed here in terms of a generalization of euler s elastica theory the model includes a linear stress strain relation but nonlinear geometric terms properties of the rod may vary along its length and various boundary conditions are considered a rotational inertia term that is neglected in many theories is retained and is essential to the analysis by use of the equivalence of an energy and a sobolev norm and by reformulation of the equations as a semilinear system global existence of solutions is proved for any smooth initial data equilibrium solutions that are stable in the static sense of minimizing the potential energy are then proved to be stable in the dynamic sense due to liapounov

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

nonlinear dynamics of filaments i dynamical instabilities abstract the kirchhoff model provides a well established mathematical framework to study both computationaly and theoretically the dynamics of thin filaments within the approximations of linear elasticity theory the study of static solutions to these equations has a long history and the usual approach to describing their instabilities is to study the time dependent version of the kirchhoff model in the euler angle frame here we study the linear stability of the full time independent equations by introducing a new are length preserving perturbation scheme as an application we consider the instabilities of various stationary solutions such as the planar ring and straight rod subjected to twisting perturbations this scheme gives a direct proof of the existence of dynamical instabilities and provides the selection mechanism for the shape of unstable filaments

on the spectral radius of unicyclic graphs with perfect matchings let u 2k be the set of all unicyclic graphs on 2k k 2 vertices with perfect matchings let u12k be the graph on 2k vertices obtained from c3 by attaching a pendant edge and k 2 paths of length 2 at one vertex of c3 let u22k be the graph on 2k vertices obtained from c3 by adding a pendant edge at each vertex together with k 3 paths of length 2 at one of three vertices in this paper we prove that u12k and u22k have the largest and the second largest spectral radius among the graphs in u 2k when k 3

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

l percolations of complex networks given a complex network its l paths correspond to sequences of l 1 distinct nodes connected through l distinct edges the l conditional expansion of a complex network can be obtained by connecting all its pairs of nodes which are linked through at least one l path and the respective conditional l expansion of the original network is defined as the intersection between the original network and its l expansion such expansions are verified to act as filters enhancing the network connectivity consequently contributing to the identification of communities in small world models it is shown in this paper for l 2 and 3 in both analytical and experimental fashions that an evolving complex network with a fixed number of nodes undergoes successive phase transitions the so called l percolations giving rise to eulerian giant clusters it is also shown that the critical values of such percolations are a function of the network size and that the network percolates for l 3 before l 2

conformations of linear dna we examine the conformations of a model for underwound and overwound dna the molecule is represented as a cylindrically symmetric elastic string subjected to a stretching force and to constraints corresponding to a specification of the link number we derive a fundamental relation between the euler angles that describe the curve and the topological linking number analytical expressions for the spatial configurations of the molecule in the infinite length limit are obtained a unique configuration minimizes the energy for a given set of physical conditions an elastic model incorporating thermal fluctuations provides excellent agreement with experimental results on the plectonemic transition

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

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 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

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

subgraphs in random networks understanding the subgraph distribution in random networks is important for modeling complex systems in classic erdos networks which exhibit a poissonian degree distribution the number of appearances of a subgraph g with n nodes and g edges scales with network size as approximately n n g however many natural networks have a non poissonian degree distribution here we present approximate equations for the average number of subgraphs in an ensemble of random sparse directed networks characterized by an arbitrary degree sequence we find scaling rules for the commonly occurring case of directed scale free networks in which the outgoing degree distribution scales as p k approximately k gamma considering the power exponent of the degree distribution gamma as a control parameter we show that random networks exhibit transitions between three regimes in each regime the subgraph number of appearances follows a different scaling law approximately nalpha where alpha n g s 1 for gamma gamma c where s is the maximal outdegree in the subgraph and gamma c s 1 we find that certain subgraphs appear much more frequently than in erdos networks these results are in very good agreement with numerical simulations this has implications for detecting network motifs subgraphs that occur in natural networks significantly more than in their randomized counterparts

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