the spectral radius of tricyclic graphs with n vertices and k pendent vertices in this paper we determine graphs with the largest spectral radius among all the tricyclic graphs with n vertices and k pendent vertices

on the largest kth eigenvalues of trees abstract we consider the only remaining unsolved case n 0 mod k for the largest k th eigenvalue of trees with n vertices we give complete solutions for the cases k 2 3 4 5 and give some necessary conditions for extremal trees in general cases

on growth rates of subadditive functions for semiflows abstract let x t x be a semiflow on a compact metric space x a function f x t x is subadditive with respect to if f x t s f x t f x t nbsp s we define the maximal growth rate of f to be sup x x lim sup t 1 t f x t this growth rate is shown to equal the maximal growth rate of the subadditive function restricted to the minimal center of attraction of the semiflow applications to birkhoff sums characteristic exponents of linear skew product semiflows on banach bundles and average lyapunov functions are developed in particular a relationship between the dynamical spectrum and the measurable spectrum of a linear skew product flow established by r a johnson k j palmer and g r sell siam j math anal 18 1987 1 33 is extended to semiflows in an infinite dimensional setting

on the spectral radius of trees with fixed diameter let t n d be the set of trees on n vertices with diameter d in this paper the first d2 1 spectral radii of trees in the set t n d 3 d n 4 are characterized

proof of a conjecture on the spectral radius of c4 free graphs abstract let g be a graph with n vertices and be the spectral radius of its adjacency matrix recently nikiforov showed that if g has no 4 cycle then 2 n 1 0 with equality if and only if g is the friendship graph however this bound is not attainable when n is even he conjectured that if g is a c 4 free graph with even number of vertices then 3 2 n 1 1 0 with equality if and only if g is a star of order n with n 2 1 disjoint additional edges we prove the conjecture in this paper

w tsv weighted topological signature vector for lexicon reduction in handwritten arabic documents this paper proposes a holistic lexicon reduction method for ancient and modern handwritten arabic documents the word shape is represented by the weighted topological signature vector w tsv which encodes graph data into a low dimensional vector space three directed acyclic graph dag representations are proposed for arabic word shapes based on topological and geometrical features lexicon reduction is achieved by a nearest neighbors search in the w tsv space the proposed framework has been tested on the ifn enit and the ibn sina databases achieving respectively a degree of reduction of 83 5 and 92 9 for an accuracy of reduction of 90

the kth largest eigenvalue of a tree abstract let k t be the k th eigenvalue of a tree x the largest integer not greater than x it is shown that a tree with n vertices has k t n 2 k for 2 k n 2 and this upper bound is best possible for n 1 mod k

ordering of some uniform supertrees with larger spectral radii abstract a supertree is a connected and acyclic hypergraph we study some uniform supertrees with larger spectral radii we first define a new type of edge moving operation on hypergraphs and study its applications on the comparison of the spectral radii of hypergraphs by using this new operation on some supertrees together with the general edge moving operation introduced by li shao and qi in 2015 and using a result by zhou et al in 2014 about the relation between the spectral radii of an ordinary graph and its power hypergraph we are able to determine the first eight k uniform supertrees on n vertices with the largest spectral radii

the spectral radius of unicyclic and bicyclic graphs with n vertices and k pendant vertices in this paper we determine graphs with the largest spectral radius among all the unicyclic and all the bicyclic graphs with n vertices and k pendant vertices respectively

on the least eigenvalue of cacti with pendant vertices abstract a connected graph is a cactus if any two of its cycles share at most one common vertex we determine the unique graph with minimum least eigenvalue among the cacti with fixed numbers of vertices and pendant vertices

robust permanence for interacting structured populations abstract the dynamics of interacting structured populations can be modeled by d x i d t a i x x i where x i r n i x x 1 x k and a i x are matrices with non negative off diagonal entries these models are permanent if there exists a positive global attractor and are robustly permanent if they remain permanent following perturbations of a i x necessary and sufficient conditions for robust permanence are derived using dominant lyapunov exponents i of the a i x with respect to invariant measures the necessary condition requires max i i 0 for all ergodic measures with support in the boundary of the non negative cone the sufficient condition requires that the boundary admits a morse decomposition such that max i i 0 for all invariant measures supported by a component of the morse decomposition when the morse components are axiom a uniquely ergodic or support all but one population the necessary and sufficient conditions are equivalent applications to spatial ecology epidemiology and gene networks are given

maximizing the spectral radius of graphs with fixed minimum degree and edge connectivity abstract the spectral radius g of a graph g is the largest eigenvalue of the adjacency matrix a g suppose a graph g 0 maximizes the spectral radius over the class of graphs of order n with fixed minimum degree and edge connectivity in this paper we mainly show that g 0 b n where b n is obtained by adding edges between k 1 and k n 1 a property of the adjacency matrix of g 0 is also obtained moreover graphs that maximize g over the class of graphs with minimum degree and edge connectivity for 0 1 2 3 are completely determined

limit cycles for competitor competitor mutualist lotka volterra systems abstract it is known that a limit cycle or periodic coexistence can occur in a competitor competitor mutualist lotka volterra system x 1 x 1 r 1 a 11 x 1 a 12 x 2 a 13 x 3 x 2 x 2 r 2 a 21 x 1 a 22 x 2 a 23 x 3 x 3 x 3 r 3 a 31 x 1 a 32 x 2 a 33 x 3 where r i a i j are positive real constants x liang j jiang the dynamical behavior of type k competitive kolmogorov systems and its applications to 3 dimensional type k competitive lotka volterra systems nonlinearity 16 2003 785 801 in this paper we shall construct an example with at least two limit cycles and furthermore we will show that the number of periodic orbits and hence a fortiori of limit cycles is finite it is also shown that contrary to three dimensional competitive lotka volterra systems nontrivial periodic coexistence does occur even if none of the three species can resist invasion from either of the others in this case new amenable conditions are given on the coefficients under which the system has no nontrivial periodic orbits these conditions imply that the positive equilibrium if it exists is globally asymptotically stable

further results on the least eigenvalue of connected graphs in this paper we identify within connected graphs of order n and size n k with 0 k 4 and n k 5 the graphs whose least eigenvalue is minimal it is also observed that the same graphs have the largest spectral spread if n is large enough

the spectral radius of trees on k pendant vertices abstract in this paper we consider the following problem of all trees of order n with k pendant vertices n k fixed which achieves the maximal spectral radius we show that the maximal spectral radius is obtained uniquely at t n k where t n k is a tree obtained from a star k 1 k and k paths of almost equal lengths by joining each pendant vertex of k 1 k to an end vertex of one path we also discuss the spectral radius of t n k and get some results

the graphs with spectral radius between 2 and 2 5 abstract we complete the determination of the graphs in the title begun by cvetkovic doob and gutman

graphs with least eigenvalue 2 ten years on abstract the authors monograph spectral generalizations of line graphs was published in 2004 following the successful use of star complements to complete the classification of graphs with least eigenvalue 2 guided by citations of the book we survey progress in this area over the past decade some new observations are included

coexistence for species sharing a predator a class of equations describing the dynamics of two prey sharing a common predator are considered even though the boundary and internal dynamics can exhibit oscillatory behavior it is shown these equations are permanent if only if they admit a positive equilibrium going beyond permanence a subclass of equations are constructed that are almost surely permanent but not permanent there exists an attractor in the positive orthant that attracts lebesgue almost every but not every initial condition

coxeter energy of graphs abstract we study the concept of the coxeter energy of graphs and digraphs quivers as an analogue of gutman s adjacency energy which has applications in theoretical chemistry and is a recently widely investigated graph invariant coxeter energy ce g of a di graph g is defined to be the sum of the absolute values of all complex eigenvalues of the coxeter matrix associated with g our main inspiration for the study comes from the coxeter formalism appearing in group theory lie theory representation theory of algebras mathematical physics and other contexts we focus on the coxeter energy of trees and we prove that the path resp the maximal star has the smallest resp the greatest coxeter energy among all trees resp two large subclasses of trees with fixed number of vertices we provide several other related results as the characterization of trees with second smallest and second greatest coxeter energy bounds for coxeter energy and general facts on coxeter spectra of graphs extending known results e g for salem trees and for certain special real coxeter eigenvalues of trees additionally we discuss few other energy like quantities for the coxeter spectra of di graphs including coxeter energy normalized by the trace of the coxeter matrix and the quantities derived from variants of coulson integral formula

a study of eigenspaces of graphs abstract we investigate the relationship between the structure of a graph and its eigenspaces the angles between the eigenspaces and the vectors of the standard basis of r n play an important role the key notion is that of a special basis for an eigenspace called a star basis star bases enable us to define a canonical bases of r n associated with a graph and to formulate an algorithm for graph isomorphism

maximal reflexive cacti with four cycles the approach via smith graphs abstract cacti or treelike graphs are graphs whose all cycles are mutually edge disjoint graphs with the property 2 2 are called reflexive graphs where 2 is the second largest eigenvalue of the corresponding 0 1 adjacency matrix the property 2 2 is a hereditary one i e all induced subgraphs of a reflexive graph are also reflexive this is why we represent reflexive graphs through the maximal graphs within a given class such as connected cacti with a fixed number of cycles in previous work we have determined all maximal reflexive cacti with four cycles whose cycles do not form a bundle and pointed out the role of so called pouring of smith graphs in their construction in this paper besides pouring we show several other patterns of the appearance of smith trees in those constructions these include splitting of a smith tree adding an edge to a smith tree and then splitting of the resulting graph identifying two vertices of a smith tree and then splitting the resulting graph our results show that the presence of smith trees is evident in all such maximal reflexive cacti with four cycles and that in most of them smith graphs appear in the described way

numerical stability of fast computation algorithms of zernike moments abstract a detailed comparative study of the numerical stability of the recursive algorithms widely used to calculate the zernike moments of an image is presented in this paper while many papers introducing fast algorithms for the computation of zernike moments have been presented in the literature there is not any work studying the numerical behaviour of these methods these algorithms have been in the past compared to each other only according to their computational complexity without been given the appropriate attention as far as their numerical stability is concerned being the most significant part of the algorithms reliability the present contribution attempts to fill this gap in the literature since it mainly demonstrates that the usefulness of a recursive algorithm is defined not only by its low computational complexity but most of all by its numerical robustness this paper exhaustively compares some well known recursive algorithms for the computation of zernike moments and sets the appropriate conditions in which each algorithm may fall in an unstable state the experiments show that any of these algorithms can be unstable under some conditions and thus the need to develop more stable algorithms is of major importance

on growth rates of sub additive functions for semi flows determined and random cases abstract let r m m be a measurable random dynamical systems on the compact metric space m over f p t t r with time r let m p and e p denote the set of all invariant measures on m and the set of all ergodic invariant measures whose marginal on coincide with p respectively a function f r m r is sub additive with respect to if f t s x f t x f s t t x we define the maximal growth rate of f to be lim sup t 1 t sup x m f t x for p a e it is shown that it is equal to max m p where lim t 1 t m f t x d and there exists e p such that max m p the result may have some applications in the study of the dynamical spectrum of infinite dimension random dynamical systems and robust permanence for differential equations

on the spectral radius of graphs with cut edges we study the spectral radius of graphs with n vertices and k cut edges in this paper we show that of all the connected graphs with n vertices and k cut edges the maximal spectral radius is obtained uniquely at k k where k k n is a graph obtained by joining k independent vertices to one vertex of kn k we also discuss the limit point of the maximal spectral radius

the second largest eigenvalue of a tree denote by 2 t the second largest eigenvalue of a tree t an easy algorithm is given to decide whether 2 t for a given number and a structure theorem for trees with 2 t is proved also it is shown that a tree t with n vertices has 2 t lsqb n 3 2rsqb12 this bound is best possible for odd n

criteria for cr robust permanence abstract let xi xifi x i 1 n be a cr vector field that generates a dissipative flow on the positive cone of rn is called permanent if the boundary of the positive cone is repelling is called crrobustly permanent if remains permanent for sufficiently small cr perturbations of the vector field a necessary condition and a sufficient condition for cr robust permanence involving the average per capita growth rates fi d with respect to invariant measures are derived the necessary condition requires that inf maxi fi d 0 where the infimum is taken over ergodic measures with compact support in the boundary of the positive cone the sufficient condition requires that the boundary flow admit a morse decomposition m1 mk such that every mj satisfies min maxi fi d 0 where the minimum is taken over invariant measures with support in mj as applications we provide a sufficient condition for cr robust permanence of lotka volterra models and a topological characterization of cr robust permanence for food chain models

robust uniform persistence in discrete and continuous nonautonomous systems abstract this is an extension of the work in salceanu 2011 14 to nonautonomous systems of difference and differential equations on the positive cone of r m that exhibit a positively invariant boundary hyperplane x it is shown that when a compact subset of x which attracts all orbits in x is a robust uniform weak repeller robust uniform persistence for the complementary dynamics i e the dynamics in r m x is obtained additional assumptions are made to deal with the nonautonomous nature of the systems some particular cases that often occur in applications are discussed and then sufficient conditions for the robust uniform persistence of the disease in two epidemic models from the literature are given

an exact recursion for the composite nearest neighbor degeneracy for a 2 n lattice space a set theoretic argument is utilized to develop a recursion relation that yields exactly the composite nearest neighbor degeneracy for simple indistinguishable particles distributed on a 2 n lattice space the associated generating functions as well as the expectation of the resulting statistics are also treated

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

on the spectral radii and the signless laplacian spectral radii of c cyclic graphs with fixed maximum degree if g is a connected undirected simple graph on n vertices and n c 1 edges then g is called a c cyclic graph specially g is called a tricyclic graph if c 3 let g be the maximum degree of g in this paper we determine the structural characterizations of the c cyclic graphs which have the maximum spectral radii resp signless laplacian spectral radii in the class of c cyclic graphs on n vertices with fixed maximum degree n c 12 moreover we prove that the spectral radius of a tricyclic graph g strictly increases with its maximum degree when g 1 6 2n32 and identify the first six largest spectral radii and the corresponding graphs in the class of tricyclic graphs on n vertices

the signless laplacian spectral radius of tricyclic graphs and trees with k pendant vertices abstract in this paper we consider the following problem of all tricyclic graphs or trees of order n with k pendant vertices n k fixed which achieves the maximal signless laplacian spectral radius we determine the graph with the largest signless laplacian spectral radius among all tricyclic graphs with n vertices and k pendant vertices then we show that the maximal signless laplacian spectral radius among all trees of order n with k pendant vertices is obtained uniquely at t n k where t n k is a tree obtained from a star k 1 k and k paths of almost equal lengths by joining each pendant vertex to one end vertex of one path we also discuss the signless laplacian spectral radius of t n k and give some results

uniform persistence and chain recurrence soit 7b e d un espace metrique localement compact et e un sous ensemble clos de 7b e soit un systeme dynamique sur e on reconsidere la notion de persistance uniforme dans le cadre de la theorie de conley des ensembles invariants

a 1 0 1 and sparsest basis for the null space of a forest in optimal time abstract given a matrix the null space problem asks for a basis of its null space having the fewest nonzeros this problem is known to be np complete and even hard to approximate the null space of a forest is the null space of its adjacency matrix sander and sander 2005 and akbari et al 2006 independently proved that the null space of each forest admits a 1 0 1 basis we devise an algorithm for determining a sparsest basis of the null space of any given forest which in addition is a 1 0 1 basis our algorithm is time optimal in the sense that it takes time at most proportional to the number of nonzeros in any sparsest basis of the null space of the input forest moreover we show that given a forest f on n vertices the set of those vertices x for which there is a vector in the null space of f that is nonzero at x and the number of nonzeros in any sparsest basis of the null space of f can be found in o n time

the least eigenvalue of unicyclic graphs with n vertices and k pendant vertices abstract let u n k be the set of unicyclic graphs with n vertices and k pendant vertices in this paper we determine the unique graph with the minimal least eigenvalue among all graphs in u n k the work is related with that of guo s g guo the spectral radius of unicyclic and bicyclic graphs with n vertices and k pendant vertices linear algebra appl 408 2005 78 85 which determined the unicyclic graph with the maximal spectral radius in u n k we can observe that the extremal graph on the least eigenvalue is different from that on the spectral radius

some properties of lattice autoregressive filters an autoregressive filter is defined either by the components of the regression vector or by the reflection coefficients appearing in its lattice representation the mathematical expression of the regression vector in terms of the reflection coefficients is very complex but many structural properties can be obtained without this exact expression in this paper we present some examples of such structural properties and we apply these results to prove some extremal properties of stable filters such as the maximum value of the components of the regression vector or the maximum value of its norm moreover some properties of the boundary of the stability domain are discussed

on ordinary and signless laplacian spectral radius of graphs with fixed number of branch vertices the unique graphs with maximum spectral radius and signless laplacian spectral radius are determined among trees unicyclic graphs and bicyclic graphs respectively with fixed numbers of vertices and branch vertices vertices of degree at least 3 using a unified approach

indices for special classes of trees abstract in this article we study the index the largest eigenvalue of the adjacency matrix in two special classes of trees namely starlike trees and double brooms for each class we determine conditions for the index to be integer we examine conditions in order to compare the index of a starlike tree with 1 where is the maximum degree of the graph we also prove that indices of a broom like tree and a double broom except for some cases are between and 1 characterizing when occurs one equality furthermore we build in these classes infinite families of non integral trees with integer index

comparing ls fir filtering and l step ahead linear prediction this comparative study of the l step ahead linear prediction and least squares finite impulse response ls fir filtering problems emphasizes the numerical behavior of the resulting toeplitz systems it is shown that although these systems are similar the restraints on the autocorrelation coefficients fundamentally differentiate them in the process of doing so a new algorithmic scheme for the computation of the lagged lattice coefficients is developed which exhibits fundamentally improved numerical behavior moreover explicit formulas for the supremums of the absolute values of both the lagged lattice and filter coefficients are found theoretically and are experimentally confirmed by using the proposed algorithm finally the bounds of the ls fir filter coefficients are treated in comparison with the supremums of the lagged quantities

notes on the second largest eigenvalue of a graph abstract for a fixed real number r we give several necessary and or sufficient conditions for a graph to have the second largest eigenvalue of the adjacency matrix or signless laplacian matrix less then or equal to r

interlacing eigenvalues on some operations of graphs in this paper we focus on some operations of graphs and give a kind of eigenvalue interlacing in terms of the adjacency matrix standard laplacian and normalized laplacian also we explore some applications of this interlacing

spectral radius of graphs with given matching number abstract in this paper we show that of all graphs of order n with matching number the graphs with maximal spectral radius are kn if n 2 or 2 1 k 2 1 k n 2 1 if 2 2 n k k n or k 2 1 k n 2 1 if n 3 2 k k n if n 3 2 where k t is the empty graph on t vertices

kolmogorov vector fields with robustly permanent subsystems the following results are proven all subsystems of a dissipative kolmogorov vector field i xifi x are robustly permanent if and only if the external lyapunov exponents are positive for every ergodic probability measure with support in the boundary of the nonnegative orthant if the vector field is also totally competitive its carrying simplex is c1 applying these results to dissipative lotka volterra systems robust permanence of all subsystems is equivalent to every equilibrium x satisfying fi x 0 whenever xi 0 if in addition the lotka volterra system is totally competitive then its carrying simplex is c1

uniform and strict persistence in monotone skew product semiflows with applications to non autonomous nicholson systems abstract we determine sufficient conditions for uniform and strict persistence in the case of skew product semiflows generated by solutions of non autonomous families of cooperative systems of odes or delay fdes in terms of the principal spectrums of some associated linear skew product semiflows which admit a continuous separation our conditions are also necessary in the linear case we apply our results to a noncooperative almost periodic nicholson system with a patch structure whose persistence turns out to be equivalent to the persistence of the linearized system along the null solution

the stability under perturbations of repulsive sets on considere le systeme suivant sur r n x f x ou f f 1 f n x x 1 x n on demontre qu un theoreme de fonction de lyapunov inverse est verifie garantissant l existence d une fonction de lyapunov u et que l on peut choisir u de lipschitz

a conjecture on the second largest eigenvalue of a tree with perfect matchings abstract in this paper we present an upper bound for the second largest eigenvalue of a tree on n 2 k 4 t t 2 vertices with perfect matchings at the same time the few largest second largest eigenvalues over the class of trees on n 2 k 4 t t 2 vertices with perfect matchings are also obtained

on the largest eigenvalues of bipartite graphs which are nearly complete abstract for positive integers p q r s and t satisfying rt p and st q let g p q r s t be the bipartite graph with partite sets u 1 u p and v 1 v q such that u i and v j are not adjacent if and only if there exists a positive integer k with 1 k t such that k 1 r 1 i kr and k 1 s 1 j ks in this paper we study the largest eigenvalues of bipartite graphs which are nearly complete we first compute the largest eigenvalue and all other eigenvalues of g p q r s t and then list nearly complete bipartite graphs according to the magnitudes of their largest eigenvalues these results give an affirmative answer to 1 conjecture 1 2 when the number of edges of a bipartite graph with partite sets u and v having u p and v q for p q is pq 2 furthermore we refine 1 conjecture 1 2 for the case when the number of edges is at least pq p 1

indexing hierarchical structures using graph spectra hierarchical image structures are abundant in computer vision and have been used to encode part structure scale spaces and a variety of multiresolution features in this paper we describe a framework for indexing such representations that embeds the topological structure of a directed acyclic graph dag into a low dimensional vector space based on a novel spectral characterization of a dag this topological signature allows us to efficiently retrieve a promising set of candidates from a database of models using a simple nearest neighbor search we establish the insensitivity of the signature to minor perturbation of graph structure due to noise occlusion or node split merge to accommodate large scale occlusion the dag rooted at each nonleaf node of the query votes for model objects that share that part effectively accumulating local evidence in a model dag s topological subspaces we demonstrate the approach with a series of indexing experiments in the domain of view based 3d object recognition using shock graphs

ordering trees and graphs with few cycles by algebraic connectivity abstract several approaches for ordering graphs by spectral parameters are presented in the literature we can find graph orderings either by the greatest eigenvalue spectral radius or index or by the sum of the absolute values of the eigenvalues the energy of a graph or by the second smallest eigenvalue of the laplacian matrix the algebraic connectivity among others by considering the fact that the algebraic connectivity is related to the connectivity and shape of the graphs several structural properties of graphs relative to this parameter have been studied hence a large number of papers about ordering graphs by algebraic connectivity mainly about trees and graphs with few cycles have been published this paper surveys the significant results concerning these topics trying to focus on possible points to be investigated in order to understand the difficulties to obtain partial orderings via algebraic connectivity

on the spectral radius of quasi tree graphs abstract a connected graph g v e is called a quasi tree if there exists u 0 v g such that g u 0 is a tree denote q n d 0 g g is a quasi tree graph of order n with g u 0 being a tree and d g u 0 d 0 in this paper we determined the maximal and the second maximal spectral radii of all quasi tree graphs in the set q n d 0

on the spectral radius of bicyclic graphs with n vertices and diameter d bicyclic graphs are connected graphs in which the number of edges equals the number of vertices plus one in this paper we determine the graph with the largest spectral radius among all bicyclic graphs with n vertices and diameter d as an application we give first three graphs among all bicyclic graphs on n vertices ordered according to their spectral radii in decreasing order

on the spectra of certain rooted trees abstract let t be an unweighted tree with vertex root v which is the union of two trees t 1 v 1 e 1 t 2 v 2 e 2 such that v 1 v 2 v and t 1 and t 2 have the property that the vertices in each of their levels have equal degree we characterize completely the eigenvalues of the adjacency matrix and of the laplacian matrix of t they are the eigenvalues of symmetric tridiagonal matrices whose entries are given in terms of the vertex degrees moreover we give some results about the multiplicity of the eigenvalues applications to some particular trees are developed

a robust parallelizable o m a posteriori recursive least squares algorithm for efficient adaptive filtering this article presents a new recursive least squares rls adaptive algorithm the proposed computational scheme uses a finite window by means of a lemma for the system matrix inversion that is for the first time stated and proven here the new algorithm has excellent tracking capabilities moreover its particular structure allows for stabilization by means of a quite simple method its stabilized version performs very well not only for a white noise input but also for nonstationary inputs as well it is shown to follow music speech environmental noise etc with particularly good tracking properties the new algorithm can be parallelized via a simple technique its parallel form is very fast when implemented with four processors

ordering trees by their largest eigenvalues abstract the set of trees with n vertices is denoted by t n hofmeister has determined the first five values of the largest eigenvalue of trees in t n and the corresponding trees for these values linear algebra appl 260 1997 43 in other words an order of the first five trees in t n by their largest eigenvalues has been given focus on the same purpose we shall give a partition for trees in t n first in the this paper and then extend this order to the eighth tree

ordering trees by their largest eigenvalues abstract let t and 1 t denote the maximum degree and the largest eigenvalue of a tree t respectively let t n be the set of trees on n vertices and t n t t n t in the present paper among the trees in t n n 4 we characterize the tree which alone minimizes the largest eigenvalue as well as the tree which alone maximizes the largest eigenvalue when n 2 2 n 1 furthermore it is proved that for two trees t1 and t2 in t n n 4 if t 1 2 n 3 1 and t1 t2 then 1 t1 1 t2 by applying this result we extend the order of trees in t n by their largest eigenvalues to the 13th tree when n 12 this extends the results of hofmeister linear algebra appl 260 1997 43 and chang et al linear algebra appl 370 2003 175

a note on the spectral characterization of strongly connected bicyclic digraphs let d be a digraph with vertex set v d and a be the adjacency matrix of d in this paper we characterize the extremal digraphs which achieve the maximum and minimum spectral radius among strongly connected bicyclic digraphs furthermore we show that any strongly connected bicyclic digraph is determined by the spectrum

smoothness of the carrying simplex for discrete time competitive dynamical systems a characterization of neat embedding abstract the paper is concerned with the question of smoothness of the carrying simplex s for a discrete time dissipative competitive dynamical system we give a necessary and sufficient criterion for s being a c 1 submanifold with corners neatly embedded in the nonnegative orthant formulated in terms of inequalities between lyapunov exponents for ergodic measures supported on the boundary of the orthant this completes one thread of investigation occasioned by a question posed by m w hirsch in 1988 besides amenable conditions are presented to guarantee the c r r 1 smoothness of s in the time periodic competitive kolmogorov systems of odes examples are also presented one in which s is of class c 1 but not neatly embedded the other in which s is not of class c 1

graphs for which the least eigenvalue is minimal ii let g be a connected graph whose least eigenvalue g is minimal among the connected graphs of prescribed order and size we show first that either g is complete or g is a simple eigenvalue in the latter case the sign pattern of a corresponding eigenvector determines a partition of the vertex set and we study the structure of g in terms of this partition we find that g is either bipartite or the join of two graphs of a simple form

laplacian coefficients of trees with given number of leaves or vertices of degree two abstract let g be a simple undirected n vertex graph with the characteristic polynomial of its laplacian matrix l g det i l g k 0 n 1 k c k n k it is well known that for trees the laplacian coefficient c n 2 is equal to the wiener index of g while c n 3 is equal to the modified hyper wiener index of graph using a result of zhou and gutman on the relation between the laplacian coefficients and the matching numbers in subdivided bipartite graphs we characterize the trees with k leaves pendent vertices which simultaneously minimize all laplacian coefficients in particular this extremal balanced starlike tree s n k minimizes the wiener index the modified hyper wiener index and recently introduced laplacian like energy we prove that graph s n n 1 p has minimal laplacian coefficients among n vertex trees with p vertices of degree two in conclusion we illustrate on examples of these spectrum based invariants that the opposite problem of simultaneously maximizing all laplacian coefficients has no solution and pose a conjecture on extremal unicyclic graphs with k leaves

on the two largest eigenvalues of trees abstract very little is known about upper bounds for the largest eigenvalues of a tree that depend only on the vertex number starting from a classical upper bound for the largest eigenvalue some refinements can be obtained by successively removing trees from consideration the results can be used to characterize those trees that maximize the second largest eigenvalue this corrects a result from the literature and it includes a proof of a conjecture of neumaier the main tool for this endeavor is the theory of partial engenvectors

new method and new results on the order of spectral radius this paper presents a simple method to order trees unicyclic graphs and bicyclic graphs according to their spectral radii which is based on the application of the majorization theorem of tree unicyclic and bicyclic graphs respectively moreover we obtain some new results on the order of spectral radii of unicyclic and bicyclic graphs by employing this new method

bounds on graph eigenvalues abstract upper and lower estimates are found for the maximum of the kth eigenvalue of a graph as a function of the number of vertices or edges

sharp upper bounds on the second largest eigenvalues of connected graphs abstract let 2 be the second largest eigenvalue of a graph powers 1988 4 gave some upper bounds of 2 for general graphs and bipartite graphs respectively considering that these bounds are not always attainable for connected graphs we present sharp upper bounds of 2 for connected graphs and connected bipartite graphs in this paper moreover the extremal graphs are completely characterized

on graphs whose signless laplacian index does not exceed 4 5 abstract let a g and d g be respectively the adjacency matrix and the degree matrix of a graph g the signless laplacian matrix of g is defined as q g d g a g the q spectrum of g is the set of the eigenvalues together with their multiplicities of q g the q index of g is the maximum eigenvalue of q g the possibilities for developing a spectral theory of graphs based on the signless laplacian matrices were discussed by cvetkovic et al d cvetkovic p rowlinson s k simic signless laplacians of finite graphs linear algebra appl 423 2007 155 171 in the latter paper the authors determine the graphs whose q index is in the interval 0 4 in this paper we investigate some properties of q spectra of graphs especially for the limit points of the q index by using these results we characterize respectively the structures of graphs whose the q index lies in the intervals 4 2 5 2 5 2 and 2 4 5 where 1 3 54 6 33 1 3 54 6 33 1 3 2 382975767

on the spectral radius of unicyclic graphs with fixed diameter in the paper we will determine graphs with the maximal spectral radius among all the unicyclic graphs with n vertices and diameter d

spectral radius of digraphs with given dichromatic number abstract let d be a digraph with vertex set v d a partition of v d into k acyclic sets is called a k coloring of d the minimum integer k for which there exists a k coloring of d is the dichromatic number d of the digraph d denote g n k the set of the digraphs of order n with the dichromatic number k 2 in this note we characterize the digraph which has the maximal spectral radius in g n k our result generalizes the result of 8 by feng et al

on the fine structure of the global attractor of a uniformly persistent flow abstract we study the internal structure of the global attractor of a uniformly persistent flow we show that the restriction of the flow to the global attractor has duality properties which can be expressed in terms of certain attractor repeller decompositions we also study some natural morse decompositions of the flow and calculate their morse equations these equations provide necessary and sufficient conditions for the existence of attractors with the shape of s 1 or such that their suspension has spherical shape

uniform persistence for nonautonomous and random parabolic kolmogorov systems abstract the purpose of this paper is to investigate uniform persistence for nonautonomous and random parabolic kolmogorov systems via the skew product semiflows approach it is first shown that the uniform persistence of the skew product semiflow associated with a nonautonomous random parabolic kolmogorov system implies that of the system various sufficient conditions in terms of the so called unsaturatedness and or lyapunov exponents for uniform persistence of the skew product semiflows are then provided among others it is shown that if the associated skew product semiflow has a global attractor and its restriction to the boundary of the state space has a morse decomposition which is unsaturated or whose external lyapunov exponents are positive then it is uniformly persistent more specific conditions are discussed for uniform persistence in n species particularly 3 species random competitive systems

on thec1stability conjecture for flows we give a proof of thec1stability conjecture of palis and smale for flows which has reduced to proving thatc1structural stability implies axiom a the proof is based on the fundamental work of liao and mane and on the recent powerfulc1connecting lemma of hayashi

on the spectral radius of trees abstract in this paper we investigated the spectral radius of trees and obtained the following result let t be a tree on n vertices let m t denote one maximum matching of t and m t i let t i be a tree as shown in fig 1 then t 1 2 n i 1 n i 1 2 4 n 2i 1 and equality holds if and only if t t i

persistence in dynamical systems soit un espace metrique localement compact de metrique d et soit e un sous ensemble clos de a frontiere e et interieur e on suppose que l on a un flot continu 7b f defini sur e tel que e est invariant par 7b f on considere la notion de persistance et on donne un critere de persistance

the minimum spectral radius of graphs with a given independence number abstract the independence number g of g is defined as the maximum cardinality of a set of pairwise non adjacent vertices which is called an independent set in this paper we characterize the graphs which have the minimum spectral radius among all the connected graphs of order n with independence number 1 2 n 2 n 2 1 n 3 n 2 n 1

uniform persistence and hopf bifurcations in r n we consider parameterized families of flows in locally compact metrizable spaces and give a characterization of those parameterized families of flows for which uniform persistence continues on the other hand we study the generalized poincare andronov hopf bifurcations of parameterized families of flows at boundary points of r n or more generally of an n dimensional manifold and show that this kind of bifurcations produce a whole family of attractors evolving from the bifurcation point and having interesting topological properties in particular in some cases the bifurcation transforms a system with extreme non permanence properties into a uniformly persistent one we study in the paper when this phenomenon happens and provide an example constructed by combining a holling type interaction with a pitchfork bifurcation