analytic invariant manifolds for sequences of diffeomorphisms abstract we obtain real analytic invariant manifolds for trajectories of maps assuming only the existence of a nonuniform exponential behavior we also consider the more general case of sequences of maps which corresponds to a nonautonomous dynamics with discrete time we emphasize that the maps that we consider are defined in a real euclidean space and thus one is not able to obtain the invariant manifolds from a corresponding procedure to that in the nonuniform hyperbolicity theory in the context of holomorphic dynamics we establish the existence both of stable and unstable manifolds and of center manifolds as a byproduct of our approach we obtain an exponential control not only for the trajectories on the invariant manifolds but also for all their derivatives

the kirchhoff indices and the matching numbers of unicyclic graphs the kirchhoff index of a connected graph is the sum of resistance distances between all unordered pairs of vertices in the graph in this paper we determine the minimum kirchhoff index among the unicyclic graphs with fixed number of vertices and matching number and characterize the extremal graphs

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

some rules on resistance distance with applications in this work some rules for resistance distances of a graph g are established let s be a set of vertices of g such that all vertices in s have the same neighborhood n in g s if s 2 3 4 simple formulae are derived to compute resistance distances between vertices in s in terms of the cardinality of n these show that resistance distances between vertices in s depend only on the cardinality of n and the induced subgraph g s one question arises naturally does this property hold for s with arbitrarily many vertices we answer this question by the following reduction principle resistance distances between vertices in s can be computed as in the subgraph obtained from g s n by deleting all the edges between vertices in n

full factors and co amenable inclusions we show that if m is a full factor and n subset m is a co amenable subfactor with expectation then n is also full this answers a question of popa from 1986 we also generalize a theorem of tomatsu by showing that if m is a full factor and sigma g curvearrowright m is an outer action of a compact group g then sigma is automatically minimal and m g is a full factor which has w spectral gap in m finally in the appendix we give a proof of the fact that several natural notions of co amenability for an inclusion n subset m of von neumann algebras are equivalent thus closing the cycle of implications given in anantharaman delaroche s paper in 1995

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

orbit equivalence rigidity consider a countable group acting ergodically by measure p reserving transformations on a probability space x and let r be the corresponding orbit equivalence relation on x the following rigidity phenomenon is shown there exist group actions such that the equivalence relation r on x determines the group and the action x uniquely up to finite groups the natural action of sln z on the n torus r n z n for n 2 is one of such examples the interpretation of these results in the context of von neumann algebras provides some support to the conjecture of connes on rigidity of group algebras for groups with property t our rigidity results also give examples of countable equivalence relations of type ii1 which cannot be generated mod 0 by a free action of any group this gives a negative answer to a long standing problem of feldman and moore

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

complete characterization of bicyclic graphs with the maximum and second maximum degree kirchhoff index abstract the degree kirchhoff index or multiplicative degree kirchhoff index of a connected simple graph g is defined as s g u v v g d g u d g v r g u v where dg u is the degree of a vertex u in g and rg u v is the resistance distance between the vertices u and v in this paper we completely characterize the bicyclic graphs of order n 6 having the maximum degree kirchhoff index moreover the bicyclic graphs of order n 7 with the second maximum degree kirchhoff index have also been determined

theoretical and computational methods to minimize kirchhoff index of graphs with a given edge k partiteness abstract let g be a connected graph the edge k partiteness of g is the minimum number of edges whose deletion from g yields a k partite graph the kirchhoff index kf g of g is the sum of the resistance distance between all unordered pairs of vertices in this paper we study the problem to determine the minimum kirchhoff index of graphs with a given edge k partiteness first we theoretically characterize the graphs with minimum kirchhoff index in this graph family when the edge k partiteness is not big compared to the number of vertices of g then we propose an exhaustive search algorithm to find the optimal graphs at last three strategies are used to reduce the computation complexity of our algorithm and several performance comparisons of these strategies are given

on the normalized laplacian spectral radius laplacian incidence energy and kemeny s constant abstract let g v e be a simple connected graph of order n and size m and let 1 2 n 1 n 0 be the normalized laplacian eigenvalues of g the greatest eigenvalue 1 is referred to as the normalized laplacian spectral radius the laplacian incidence energy the randic energy and kemeny s constant of g are defined by l i e g i 1 n 1 i r e g i 1 n i 1 and k g i 1 n 1 1 i respectively new lower bounds for the normalized laplacian spectral radius are obtained the acquired bounds are then used to obtain new bounds for the l i e g in addition some relations between l i e g k g r e g are determined

some unique group measure space decomposition results using an approach emerging from the theory of closable deriva tions on von neumann algebras we exhibit a class of groups cr satisfying the following property given any groups 1 2 2 cr then any free ergodic measure preserving action on a probability space 1 2 y x gives rise to a von neumann algebra with unique group measure space cartan subalgebra pairing this result with popa s orbit equivalence superrigidity theorem we obtain new examples of w superrigid actions

kirchhoff indexes of a network in this work we define the effective resistance between any pair of vertices with respect to a value 0 and a weight on the vertex set this allows us to consider a generalization of the kirchhoff index of a finite network it turns out that is the lowest eigenvalue of a suitable semi definite positive schrodinger operator and is the associated eigenfunction we obtain the relation between the effective resistance and hence between the kirchhoff index with respect to and and the eigenvalues of the associated schrodinger operator however our main aim in this work is to get explicit expressions of the above parameters in terms of equilibrium measures of the network from these expressions we derive a full generalization of foster s formulae that incorporate a positive probability of remaining in each vertex in every step of a random walk finally we compute the effective resistances and the generalized kirchhoff index with respect to a and for some families of networks with symmetries specifically for weighted wagon wheels and circular ladders

some oe and w rigidity results for actions by wreath product groups we use deformation rigidity theory in the von neumann algebra framework to study probability measure preserving actions by wreath product groups in particular we single out large families of wreath product groups satisfying various types of orbit equivalence oe rigidity for instance we show that whenever h k are icc property t groups such that h and k admit stably orbit equivalent action and such that h and k are ergodic then automatically is stably orbit equivalent to and h is stably orbit equivalent to k rigidity results for von neumann algebras arising from certain actions of such groups i e w rigidity results are also obtained

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

on the minimum kirchhoff index of graphs with a given vertex k partiteness and edge k partiteness abstract the kirchhoff index of a connected graph is the sum of the resistance distance between all unordered pairs of vertices and may also be expressed by its laplacian eigenvalues the vertex resp edge k partiteness of a graph g with n vertices is the minimum number of vertices resp edges whose deletion from g yields a k partite graph in this paper we determine the minimum kirchhoff index of graphs with a given vertex k partiteness and the minimum kirchhoff index of graphs with a given edge bipartiteness when the given edge bipartiteness is no more than n 4

orbit equivalence rigidity and bounded cohomology we establish new results and introduce new methods in the theory of measurable orbit equivalence using bounded cohomology of group representations our rigidity statements hold for a wide uncountable class of groups arising from negative curvature geometry amongst our applications are a measurable mostow type rigidity theorems for products of negatively curved groups b prime factorization results for measure equivalence c superrigidity for orbit equivalence d the first examples of continua of type ii1 equivalence relations with trivial outer automorphism group that are mutually not stably isomorphic

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

a converse to dye s theorem every non amenable countable group induces orbit inequivalent ergodic equivalence relations on standard borel probability spaces not every free ergodic measure preserving action of f 2 on a standard borel probability space is orbit equivalent to an action of a countable group on an inverse limit of finite spaces there is a treeable non hyperfinite borel equivalence relation which is not universal for treeable in the b ordering

w superrigidity for bernoulli actions of property t groups we consider group measure space 1 factors m l x arising from bernoulli actions of icc property t groups more generally of groups containing an infinite normal subgroup with the relative property t and prove a rigidity result for homomorphisms m overbar m we deduce that the action x is w superrigid i e if y is any free ergodic measure preserving action such that the factors m l x and m l y are isomorphic then the actions x and y must be conjugate moreover we show that if p m 1 is a projection then pmp does not admit a group measure space decomposition nor a group von neumann algebra decomposition the latter under the additional assumption that is torsion free we also prove a rigidity result for homomorphisms m m this time for in a larger class of groups than above now including products of non amenable groups for certain groups e g f 2 f 2 we deduce that m does not embed into pmp for any projection p m 1 and obtain a description of the endomorphism semigroup of m

on the structural theory of ii1 factors of negatively curved groups ii actions by product groups abstract this paper contains a series of structural results for von neumann algebras arising from measure preserving actions by product groups on probability spaces expanding upon the methods used earlier by the first two authors we obtain new examples of strongly solid factors as well as von neumann algebras with unique or no cartan subalgebra for instance we show that every ii 1 factor associated with a weakly amenable group in the class s of ozawa is strongly solid there is also the following product version of this result any maximal abelian subalgebra of any ii 1 factor associated with a finite product of weakly amenable groups in the class s of ozawa has an amenable normalizing algebra finally pairing some of these results with a cocycle superrigidity result of ioana it follows that compact actions by finite products of lattices in s p n 1 n 2 are virtually w superrigid

compact actions whose orbit equivalence relations are not profinite abstract let x be a measure preserving action of a countable group on a standard probability space x we prove that if the action x is not profinite and satisfies a certain spectral gap condition then there does not exist a countable to one borel homomorphism from its orbit equivalence relation to the orbit equivalence relation of any modular action i e an inverse limit of actions on countable sets as a consequence we show that if is a countable dense subgroup of a compact non profinite group g such that the left translation action g has spectral gap then g is antimodular and not orbit equivalent to any not necessarily free profinite action this provides the first such examples of compact actions partially answering a question of kechris and answering a question of tsankov

minimizing effective resistance of a graph the effective resistance between two nodes of a weighted graph is the electrical resistance seen between the nodes of a resistor network with branch conductances given by the edge weights the effective resistance comes up in many applications and fields in addition to electrical network analysis including for example markov chains and continuous time averaging networks in this paper we study the problem of allocating edge weights on a given graph in order to minimize the total effective resistance i e the sum of the resistances between all pairs of nodes we show that this is a convex optimization problem and can be solved efficiently either numerically or in some cases analytically we show that optimal allocation of the edge weights can reduce the total effective resistance of the graph compared to uniform weights by a factor that grows unboundedly with the size of the graph we show that among all graphs with n nodes the path has the largest value of optimal total effective resistance and the complete graph has the least

on the ihara zeta function and resistance distance based indices abstract we show that the ihara zeta function of a graph determines a resistance distance based invariant which is a linear combination of the kirchhoff index additive degree kirchhoff index and multiplicative degree kirchhoff index

on the generalized adjacency laplacian and signless laplacian spectra of the weighted edge corona networks abstract many problems in real world either natural or man made can be usefully represented by graphs or networks along with a complex topological structure the weight is a vital factor in characterizing some properties of real networks in this paper we define a class of the weighted edge corona product networks the generalized adjacency resp laplacian and signless laplacian spectra with two different structures are determined as applications the number of spanning trees and kirchhoff index of the weighted edge corona product networks are computed

the general randic index of trees with given number of pendent vertices the general randic index of a graph g is defined as r g u v e g d u d v where d u denotes the degree of a vertex u in g and is a real number in this paper we determine the maximum general randic indices of trees and chemical trees with n vertices and k pendent vertices for 4 k n 2 3 and 0 0 where 0 0 5122 is the unique non zero root of the equation 6 4 20 9 10 12 16 5 24 0 the corresponding extremal graphs are also characterized

bipartiteness and the least eigenvalue of signless laplacian of graphs let g be a simple graph and let b g the least eigenvalue of the signless laplacian of the graph g in this paper we focus on the relations between the least eigenvalue and some parameters reflecting the graph bipartiteness we introduce two parameters the vertex bipartiteness b g and the edge bipartiteness b g and show that b g b g b g we also define another parameter g involved with a cut set and prove that b g g g 2 g 2 where g is the maximum degree of the graph g the above two inequalities are very similar in form to those given by fiedler and mohar respectively with respect to the algebraic connectivity of laplacian of graphs which is used to characterize the connectedness of graphs

calculating the normalized laplacian spectrum and the number of spanning trees of linear pentagonal chains abstract let w n be a linear pentagonal chain with 2 n pentagons in this article according to the decomposition theorem for the normalized laplacian polynomial of w n we obtain that the normalized laplacian spectrum of w n consists of the eigenvalues of two special matrices l a of order 3 n 1 and l s of order 2 n 1 together with the relationship between the roots and coefficients of the characteristic polynomials of the above two matrices explicit closed form formulas for the degree kirchhoff index and the total number of spanning trees of w n are derived respectively finally it is interesting to find that the degree kirchhoff index of w n is approximately one half of its gutman index

measure equivalence rigidity of the mapping class group we show that the mapping class group of a compact orientable surface with higher complexity satisfies the following rigidity in the sense of measure equivalence if the mapping class group is measure equivalent to a discrete group then they are commensurable up to finite kernels moreover we describe all locally compact second countable groups containing a lattice isomorphic to the mapping class group we obtain similar results for finite direct products of mapping class groups

spectral properties of general hypergraphs abstract in this paper we investigate spectral properties of the adjacency tensor laplacian tensor and signless laplacian tensor of general hypergraphs including uniform and non uniform hypergraphs we give some properties for the spectral radius of hypergraphs and obtain spectral upper bounds for the chromatic number of hypergraphs the odd bipartiteness of hypergraphs can be recognized from the spectrum we give a relation between the analytic connectivity and edge connectivity of a hypergraph and show that a hypergraph with even rank is connected if and only if its analytic connectivity is larger than 0 we also give some relations between the analytic bipartiteness and odd bipartiteness of hypergraphs

resistance distance based graph invariants and spanning trees of graphs derived from the strong prism of a star abstract let s n 2 be the graph obtained by the strong prism of a star sn i e the strong product of k2 and sn in this paper explicit expressions for kirchhoff index multiplicative degree kirchhoff index and number of spanning tress of s n 2 are determined respectively more specially let s n r 2 be the set of subgraphs obtained by randomly deleting r vertical edges from s n 2 where 0 r n explicit formulas for kirchhoff index and number of spanning trees for any graph s n r 2 s n r 2 are established respectively moreover the kirchhoff index of s n r 2 is almost three eighths of its wiener index

on graphs with equal algebraic and vertex connectivity let g be an undirected unweighted graph on n vertices let l be its laplacian matrix and let l l i j be the group inverse of l it is known that for z l 1 2 max1 i j n s 1n li s lj s the quantity 1 z l is a lower bound on the algebraic connectivity a g of g while the vertex connectivity of g v g is an upper bound on a g we characterize the graphs g for which v g a g and subsequently prove that if n v g 2 then v g a g holds if and only if 1 z l a g v g we close with an example showing that the equality 1 z l a g does not necessarily imply that 1 z l a g v g

the domination number and the least q eigenvalue abstract in this paper for a nonbipartite graph with both order n and domination number we show that it contains a nonbipartite unicyclic spanning subgraph with the same domination number we also present some results about the changing of the domination number when the structure of a graph is changed by investigating the relation between the domination number and the least q eigenvalue of a graph we minimize the least q eigenvalue among all the nonbipartite graphs with both order n 4 and given domination number n 1 3

gromov s measure equivalence and rigidity of higher rank lattices in this paper the notion of measure equivalence me of countable groups is studied me was introduced by gromov as a measure theoretic analog of quasi isometries all lattices in the same locally compact group are measure equivalent this is one of the motivations for this notion the main result of this paper is me rigidity of higher rank lattices any countable group which is me to a lattice in a simple lie group g of higher rank is commensurable to a lattice in g

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

w superrigidity for arbitrary actions of central quotients of braid groups for any n geqslant 4 let tilde b n b n z b n be the quotient of the braid group b n through its center we prove that any free ergodic probability measure preserving pmp action tilde b n curvearrowright x mu is virtually hbox w superrigid in the following sense if l infty x rtimes tilde b n cong l infty y rtimes lambda for an arbitrary free ergodic pmp action lambda curvearrowright y nu then the actions tilde b n curvearrowright x lambda curvearrowright y are virtually conjugate moreover we prove that the same holds if tilde b n is replaced with a finite index subgroup of the direct product tilde b n 1 times cdots times tilde b n k for some n 1 ldots n k geqslant 4 the proof uses the dichotomy theorem for normalizers inside crossed products by free groups from popa and vaes 212 141 198 2014 in combination with the oe superrigidity theorem for actions of mapping class groups from kida 131 99 109 2008 similar techniques allow us to prove that if a group gamma is hyperbolic relative to a finite family of proper finitely generated residually finite infinite subgroups then the hbox ii 1 factor l infty x rtimes gamma has a unique cartan subalgebra up to unitary conjugacy for any free ergodic pmp action gamma curvearrowright x mu

integrable measure equivalence and rigidity of hyperbolic lattices we study rigidity properties of lattices in operatorname isom mathbf h n simeq mathrm so n 1 mathbb r n 3 and of surface groups in operatorname isom mathbf h 2 simeq mathrm sl 2 mathbb r in the context of integrable measure equivalence the results for lattices in operatorname isom mathbf h n n 3 are generalizations of mostow rigidity they include a cocycle version of strong rigidity and an integrable measure equivalence classification despite the lack of mostow rigidity for n 2 we show that cocompact lattices in operatorname isom mathbf h 2 allow a similar integrable measure equivalence classification

the classification problem for s local torsion free abelian groups of finite rank abstract suppose that n 2 and that s t are sets of primes then the classification problem for the s local torsion free abelian groups of rank n is borel reducible to the classification problem for the t local torsion free abelian groups of rank n if and only if s t

strong rigidity of generalized bernoulli actions and computations of their symmetry groups abstract we study equivalence relations and ii 1 factors associated with quotients of generalized bernoulli actions of kazhdan groups specific families of these actions are entirely classified up to isomorphism of ii 1 factors this yields explicit computations of outer automorphism and fundamental groups in particular every finitely presented group is concretely realized as the outer automorphism group of a continuous family of non stably isomorphic ii 1 factors

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

line star sets for laplacian eigenvalues abstract let g be a mixed graph with a nonzero laplacian eigenvalue of multiplicity k a line star set for in g is a set y of k edges of g such that is not a laplacian eigenvalue of g y it is shown that line star set exists for any nonzero laplacian eigenvalue of any mixed graph some basic properties for line star sets are given we also obtain some results on line star sets in undirected graphs

the normalized laplacians on both k triangle graph and k quadrilateral graph with their applications abstract the k triangle graph tk g is obtained from a graph g by replacing each edge in g with k 1 parallel paths in which one is of length 1 and each of the rest k paths is of length 2 whereas the k quadrilateral graph qk g is obtained from g by replacing each edge in g with k 1 parallel paths in which one is of length 1 and each of the rest k paths is of length 3 in this paper we completely determine the normalized laplacian spectrum on tk g resp qk g for any connected graph g k 2 as applications the correlation between the degree kirchhoff index the kemeny s constant and the number of spanning trees of tk g resp qk g the r th iterative k triangle graph t r k g the r th iterative k quadrilateral graph q r k g and those of g are derived our results extend those main results obtained in xie et al 2016 and li and hou 2017

balancedness and the least eigenvalue of laplacian of signed graphs abstract let g be a connected signed graph where g is the underlying simple graph and e g 1 1 is the sign function on the edges of g let l d g a be the laplacian of and 1 2 n 0 be its eigenvalues it is well known that a signed graph is balanced if and only if n 0 here we show that if is unbalanced then n estimates how much is far from being balanced in particular let resp be the frustration number resp frustration index that is the minimum number of vertices resp edges to be deleted such that the signed graph is balanced then we prove that n further we analyze the case when n in the latter setting we identify the structure of the underlying graph g and we give an algebraic condition for l which leads to the above equality

ergodic theory and maximal abelian subalgebras of the hyperfinite factor abstract let t be a free ergodic measure preserving action of an abelian group g on x the crossed product algebra rt l x g has two distinguished masas the image ct of l x and the algebra st generated by the image of g we conjecture that conjugacy of the singular masas st 1 and st 2 for weakly mixing actions t 1 and t 2 of different groups implies that the groups are isomorphic and the actions are conjugate with respect to this isomorphism our main result supporting this conjecture is that the conclusion is true under the additional assumption that the isomorphism rt 1 rt 2 such that st 1 st 2 has the property that the cartan subalgebras ct 1 and ct 2 of rt 2 are inner conjugate we discuss a stronger conjecture about the structure of the automorphism group aut rt st and a weaker one about entropy as a conjugacy invariant we study also the pukanszky and some related invariants of st and show that they have a simple interpretation in terms of the spectral theory of the action t it follows that essentially all values of the pukanszky invariant are realized by the masas st and there exist non conjugate singular masas with the same pukanszky invariant

maximizing signless laplacian or adjacency spectral radius of graphs subject to fixed connectivity in this paper we characterize the graphs with maximum signless laplacian or adjacency spectral radius among all graphs with fixed order and given vertex or edge connectivity we also discuss the minimum signless laplacian or adjacency spectral radius of graphs subject to fixed connectivity consequently we give an upper bound of signless laplacian or adjacency spectral radius of graphs in terms of connectivity in addition we confirm a conjecture of aouchiche and hansen involving adjacency spectral radius and connectivity

stable manifolds for holomorphic automorphisms we give a sucient condition for the abstract basin of attraction of a sequence of holomorphic self maps of balls in c d to be biholomorphic to c d as a consequence we get a sucient condition for the stable manifold of a point in a compact hyperbolic invariant subset of a complex manifold to be biholomorphic to a complex euclidean space our result immediately implies previous theorems obtained by jonsson varolin and by peters in particular we prove without using oseledec s theory that the stable manifold of any point where the negative lyapunov exponents are well dened is biholomorphic to a complex euclidean space our approach is based on the solution of a linear control problem in spaces of subexponential sequences and on careful estimates of the norm of the conjugacy operator by a lower triangular matrix on the space of k homogeneous polynomial endomorphisms of c d mathematics subject classication 2010 primary 37f99 secondary 32h50 37d25 37h99

a sharp lower bound for the least eigenvalue of the signless laplacian of a non bipartite graph we prove that the minimum value of the least eigenvalue of the signless laplacian of a connected non bipartite graph with a prescribed number of vertices is attained solely in the unicyclic graph obtained from a triangle by attaching a path at one of its endvertices

resistance distance in subdivision vertex join and subdivision edge join of graphs abstract the subdivision graph s g of a graph g is the graph obtained by inserting a new vertex into every edge of g let g 1 g 2 be the disjoint union of two graphs g 1 and g 2 the subdivision vertex join of g 1 and g 2 denoted by g 1 g 2 is the graph obtained from s g 1 g 2 by joining every vertex in v g 1 to every vertex in v g 2 the subdivision edge join of g 1 and g 2 denoted by g 1 g 2 is the graph obtained from s g 1 g 2 by joining every vertex in i g 1 to every vertex in v g 2 where i g 1 is the set of inserted vertices of s g 1 in this paper formulae for resistance distance in g 1 g 2 and g 1 g 2 are obtained

minimizing kirchhoff index among graphs with a given vertex bipartiteness the resistance distance between any two vertices of a graph g is defined as the effective resistance between them if each edge of g is replaced by a unit resistor the kirchhoff index kf g is the sum of the resistance distances between all the pairs of vertices in g the vertex bipartiteness vb of a graph g is the minimum number of vertices whose deletion from g results in a bipartite graph in this paper we characterize the graph having the minimum kf g values among graphs with a fixed number n of vertices and fixed vertex bipartiteness 1 v b n 3

the aα spread of a graph abstract let a g and d g be the adjacency matrix and the degree diagonal matrix of a graph g respectively for any real number 0 1 nikiforov defined the a matrix of a graph g as a g d g 1 a g the a spread of a graph is the difference between the largest eigenvalue and the smallest eigenvalue of the a matrix of the graph in this paper we obtain some new lower bounds on a spread and characterize the extremal graphs for certain cases which extend the results of a spread and q spread moreover some graph operations on a spread are presented

on the spectrum of the normalized laplacian of iterated triangulations of graphs the eigenvalues of the normalized laplacian of a graph provide information on its topological and structural characteristics and also on some relevant dynamical aspects specifically in relation to random walks in this paper we determine the spectra of the normalized laplacian of iterated triangulations of a generic simple connected graph as an application we also find closed forms for their multiplicative degree kirchhoff index kemeny s constant and number of spanning trees

continuous cocycle superrigidity for shifts and groups with one end we prove that if a finitely generated group g is not torsion then a necessary and sufficient condition for every full shift over g has continuous cocycle superrigidity is that g has one end it is a topological version of the well known popa s measurable cocycle superrigidity theorem cocycle and orbit equivalence superrigidity for malleable actions of w rigid groups invent math 170 2 243 295 2007 for the proof of sufficiency we introduce a new specification property for shifts over general groups which plays a similar role as malleable property in the measurable setting this new specification property is good enough for us to extend the method of using homoclinic equivalence relation that was introduced by schmidt pac j math 170 1 237 269 1995 to study cocycle rigidity for mathbb z d shifts indeed in this direction we prove this superrigidity result for certain more general systems and for the necessary condition we apply specker s characterization for ends of groups via the associated first cohomology groups to get the result finally combining our results with li ergod theory dyn syst arxiv 1503 01704 theorem 1 6 we have an application in continuous orbit equivalence rigidity

cocycle superrigidity and bounded cohomology for negatively curved spaces we introduce new techniques to extend superrigidity theory beyond the scope of lie or algebraic groups we construct a cohomological invariant which accounts for and generalizes all known superrigidity results for actions on negatively curved spaces together with a new vanishing result and the machinery of bounded cohomology this enables us to prove a general superrigidity theorem for actions of irreducible lattices on spaces of negative curvature we also prove a cocycle version a la zimmer

the normalized laplacian degree kirchhoff index and spanning trees of the linear polyomino chains let bn be a linear polyomino chain with n squares in this paper according to the decomposition theorem of normalized laplacian polynomial we obtain that the normalized laplacian spectrum of bn consists of the eigenvalues of two symmetric tridiagonal matrices of order n 1 together with the relationship between the roots and coefficients of the characteristic polynomials of the above two matrices explicit closed formulas of the degree kirchhoff index and the number of spanning trees of bn are derived furthermore it is interesting to find that the degree kirchhoff index of bn is approximately one half of its gutman index

on the superrigidity of malleable actions with spectral gap some of the most interesting aspects of the dynamics of measure preserving actions of countable groups on probability spaces v rx x are revealed by the study of group measure space von neumann algebras l x x t mvnl and the classification of actions up to orbit equivalence oe i e up to isomorphism of probability spaces carrying the orbits of actions onto each other although one is in von neumann algebras and the other in ergodic theory the two problems are closely related as an oe of actions v rx x a rxy has been shown to implement an algebra isomorphism l x x t l y x a taking l x onto l y and vice versa si dy fm in particular the isomorphism class of l x xi t only depends on the equivalence relation 7 r t gt 1t x g t thus von neumann equivalence vne of group actions requiring isomorphism of their group measure space algebras is weaker than oe since there are exam ples of non oe actions whose associated von neumann algebras are all isomorphic cj1 it is in general strictly weaker on the other hand oe is manifestly weaker than classical conjugacy which for free actions r rx x a rx y requires isomor phism of probability spaces a x p y v satisfying ar a 1 a so in particular r a how much weaker vne and oe can be with respect to conju gacy is best seen in the amenable case where by a celebrated theorem of connes all free ergodic actions of all infinite amenable groups give rise to the same hi factor cl and by dy ow cfw they are undistinguishable under oe as well also any embedding of algebras l x xi t c l y x a with a amenable forces t to be amenable but the non amenable case is extremely complex and for many years progress has been slow mvn2 dy me c2 cw sc even after the discovery of the first rigidity phenomena by connes in von neumann algebras c3 c4 and by zimmer in oe ergodic theory zi z2 this changed dramatically over the last 7 8 years with the advent of a variety of striking rigidity results ful gl g2 mosh h hk p1 p8 see p9 for a survey also sh for a survey on oe rigidity our aim in this paper is to investigate the most extreme such phenomena called strong rigidity which show that for certain classes of source group actions

oe and w superrigidity results for actions by surface braid groups we show that several important normal subgroups of the mapping class group of a surface satisfy the following property any free ergodic probability measure preserving action y x is stablyoe superrigid these include the central quotients of most surface braid groups and most torelli groups and johnson kernels in addition we show that all these groups satisfy the measure equivalence rigidity and we describe all their lattice embeddings using these results in combination with previous results from cik13 we deduce that any free ergodic probabilitymeasure preserving action of almost any surface braid group is stablyw superrigid i e it can be completely reconstructed from its vonneumann algebra

perturbed basins of attraction let f be an automorphism of open image in new window which has an attracting fixed point it is well known that the basin of attraction is biholomorphically equivalent to open image in new window we will show that the basin of attraction of a sequence of automorphisms f 1 f 2 is also biholomorphic to open image in new window if every f n is a small perturbation of the original map f

cocycle superrigidity for translation actions of product groups let g be either a profinite or a connected compact group and be finitely generated dense subgroups assuming that the left translation action of on g is strongly ergodic we prove that any cocycle for the left right translation action of on g with values in a countable group is virtually cohomologous to a group homomorphism moreover we prove that the same holds if g is a not necessarily compact connected simple lie group provided that contains an infinite cyclic subgroup with compact closure we derive several applications to oe and w superrigidity in particular we obtain the first examples of compact actions of f2 f2 which are w superrigid

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

new bounds for the signless laplacian spread abstract let g be an undirected simple graph the signless laplacian spread of g is defined as the maximum distance of pairs of its signless laplacian eigenvalues this paper establishes some new bounds both lower and upper for the signless laplacian spread several of these bounds depend on invariant parameters of the graph we also use a minmax principle to find several lower bounds for this spectral invariant

the least q eigenvalue with fixed domination number abstract denote by lg l the lollipop graph obtained by attaching a pendant path p v g v g 1 v g l l 1 to a cycle c v 1 v 2 v g v 1 g 3 a f g l g r a p h of order n g 1 is defined to be the graph obtained by attaching n g l pendent vertices to some of the nonpendant vertices of lg l in which each vertex other than v g l 1 is attached at most one pendant vertex a f g l graph is a f g l g r a p h in which vg is attached with pendant vertex denote by qmin the least q e i g e n v a l u e of a graph in this paper we proceed on considering the domination number the least q eigenvalue of a graph as well as their relation further results obtained are as follows i some results about the changing of the domination number under the structural perturbation of a graph are represented ii among all nonbipartite unicyclic graphs of order n with both domination number and girth g g n 1 the minimum qmin attains at a f g l graph for some l iii among the nonbipartite graphs of order n and with given domination number which contain a f g l graph as a subgraph some lower bounds for qmin are represented iv among the nonbipartite graphs of order n and with given domination number n 2

signless laplacian spectral radius and hamiltonicity we give tight conditions on the signless laplacian spectral radius of a graph for the existence of hamiltonian paths and cycles

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

weak expansiveness for actions of sofic groups in this paper we shall introduce h expansiveness and asymptotical h expansiveness for actions of sofic groups by definition each h expansive action of a sofic group is asymptotically h expansive we show that each expansive action of a sofic group is h expansive and for any given asymptotically h expansive action of a sofic group the entropy function with respect to measures is upper semi continuous and hence the system admits a measure with maximal entropy observe that asymptotically h expansive property was first introduced and studied by misiurewicz for z actions using the language of tail entropy and thus in the remaining part of the paper we shall compare our definitions of weak expansiveness for actions of sofic groups with the definitions given in the same spirit of misiurewicz s ideas when the group is amenable it turns out that these two definitions are equivalent in this setting

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

some rigidity results for non commutative bernoulli shifts abstract we introduce the outer conjugacy invariants s s s for cocycle actions of discrete groups g on type ii 1 factors n as the set of real numbers t 0 for which the amplification t of can be perturbed to an action respectively to a weakly mixing action we calculate explicitly s s s and the fundamental group of f in the case g has infinite normal subgroups with the relative property t e g when g itself has the property t of kazhdan and is an action of g on the hyperfinite ii 1 factor by connes stormer bernoulli shifts of weights t i i thus s s and f coincide with the multiplicative subgroup s of r generated by the ratios t i t j i j while s z if s 1 i e when all weights are equal and s r otherwise in fact we calculate all the 1 cohomology picture of t t 0 and classify the actions g in terms of their weights t i i in particular we show that any 1 cocycle for g vanishes modulo scalars and that two such actions are cocycle conjugate iff they are conjugate also any cocycle action obtained by reducing a bernoulli action of a group g as above on n g g m n n c tr g to the algebra pnp for p a projection in n p 0 1 cannot be perturbed to a genuine action

loewner equations on complete hyperbolic domains abstract we prove that on a complete hyperbolic domain d c q any loewner pde associated with a herglotz vector field of the form h z t z o z 2 where the eigenvalues of have strictly negative real part admits a solution given by a family of univalent mappings f t d c q which satisfies t 0 f t d c q if no real resonance occurs among the eigenvalues of then the family e t f t is uniformly bounded in a neighborhood of the origin we also give a generalization of pommerenke s univalence criterion on complete hyperbolic domains

on the sign patterns of the smallest signless laplacian eigenvector abstract let h be a connected bipartite graph whose signless laplacian matrix is q h suppose that the bipartition of h is s t and that x is the eigenvector of the smallest eigenvalue of q h it is well known that x is positive and constant on s and negative and constant on t the resilience of the sign pattern of x under addition of edges into the subgraph induced by either s or t is investigated and a number of cases in which the sign pattern of x persists are described