the effect of a graft transformation on distance spectral radius abstract we study the effect of a graft transformation on the distance spectral radius of connected graphs as applications we determine the unique n vertex non starlike tree with maximal distance spectral radius and the unique n vertex non caterpillar trees with minimal and maximal distance spectral radii and we also consider the distance spectral radius of non starlike non caterpillar trees

on the minimum eccentric distance sum of bipartite graphs with some given parameters the eccentric distance sum is a novel graph invariant with vast potential in structure activity property relationships this graph invariant displays high discriminating power with respect to both biological activity and physical properties if g vg eg is a simple connected graph then the eccentric distance sum eds of g is defined as d g v vgeg v dg v where eg v is the eccentricity of the vertex v and dg v u vgdg u v is the sum of all distances from the vertex v much extremal work has been done on trees with some given parameters by yu et al 2011 25 li et al 2012 19 and geng et al 2013 6 it is natural to consider this extremal problem on bipartite graphs with some given parameters in this paper sharp lower bound on the eds in the class of all connected bipartite graphs with a given matching number q is determined the minimum eds is realized only by the graph kq n q the extremal graph with the minimum eds in the class of all the n vertex connected bipartite graphs of odd diameter is characterized all the extremal graphs having the minimum eds in the class of all connected n vertex bipartite graphs with a given vertex connectivity are identified as well

the toda hierarchy and the kdv hierarchy this paper considers the relation between the periodic kdv hierarchy and the limit of the periodic toda hierarchies by choosing the initial data of the toda flows in a canonical way the behavior of a certain toda flow can mimic kdv flows conjecturally a method of deforming the kdv hierarchy is given

differential difference equations associated with the fractional lax operators we study integrable hierarchies associated with spectral problems of the form p q where p and q are difference operators the corresponding nonlinear differential difference equations can be viewed as inhomogeneous generalizations of the bogoyavlensky type lattices while the latter turn into the korteweg de vries equation under the continuous limit the lattices under consideration provide discrete analogs of the sawada kotera and kaup kupershmidt equations the r matrix formulation and several of the simplest explicit solutions are presented

association schemes and coding theory this paper contains a survey of association scheme theory with its algebraic and analytical aspects and of its applications to coding theory in a wide sense it is mainly concerned with a class of subjects that involve the central notion of the distance distribution of a code special emphasis is put on the linear programming method inspired by the macwilliams transform this produces upper bounds for the size of a code with a given minimum distance and lower bounds for the size of a design with a given strength the most specific results are obtained in the case where the underlying association scheme satisfies certain well defined polynomial properties this leads one into the realm of orthogonal polynomial theory in particular some universal bounds are derived for codes and designs in polynomial type association schemes throughout the paper the main concepts methods and results are illustrated by two examples that are of major significance in classical coding theory namely the hamming scheme and the johnson scheme other topics that receive special attention are spherical codes and designs and additive codes in translation schemes including z sub 4 additive binary codes

distance spectra of graphs a survey abstract in 1971 graham and pollack established a relationship between the number of negative eigenvalues of the distance matrix and the addressing problem in data communication systems they also proved that the determinant of the distance matrix of a tree is a function of the number of vertices only since then several mathematicians were interested in studying the spectral properties of the distance matrix of a connected graph computing the distance characteristic polynomial and its coefficients was the first research subject of interest thereafter the eigenvalues attracted much more attention in the present paper we report on the results related to the distance matrix of a graph and its spectral properties

asymptotic improvement of the gilbert varshamov bound on the size of binary codes given positive integers n and d let a sub 2 n d denote the maximum size of a binary code of length n and minimum distance d the well known gilbert varshamov bound asserts that a sub 2 n d spl ges 2 sup n v n d l where v n d spl sigma sub i 0 sup d sub i sup n is the volume of a hamming sphere of radius d we show that in fact there exists a positive constant c such that a sub 2 n d spl ges c2 sup n v n d 1 log sub 2 v n d 1 whenever d n spl les 0 499 the result follows by recasting the gilbert varshamov bound into a graph theoretic framework and using the fact that the corresponding graph is locally sparse generalizations and extensions of this result are briefly discussed

complete multipartite graphs are determined by their distance spectra abstract in this paper we prove that the complete multipartite graphs are determined by their distance spectra which confirms the conjecture proposed by lin hong wang and shu 2013 7 although it is well known that the complete multipartite graphs cannot be determined by their adjacency spectra

integral cayley graphs over dicyclic group abstract how to classify all the integral graphs is a challenging work as suggested by harary and schwenk in this paper we focus on one non abelian group the dicyclic group t 4 n a b a 2 n 1 a n b 2 b 1 a b a 1 and consider its corresponding cayley graphs with the help of its character table we first obtain a necessary and sufficient condition for the integrality of cayley graphs over t 4 n then we obtain several simple sufficient conditions for the integrality of cayley graphs over t 4 n in terms of the boolean algebra of a as a byproduct we determine a few infinite families of connected integral cayley graphs over t 4 n at last for a prime p we completely determine all integral cayley graphs over the dicyclic group t 4 p

two laplacians for the distance matrix of a graph abstract we introduce a laplacian and a signless laplacian for the distance matrix of a connected graph called the distance laplacian and distance signless laplacian respectively we show the equivalence between the distance signless laplacian distance laplacian and the distance spectra for the class of transmission regular graphs there is also an equivalence between the laplacian spectrum and the distance laplacian spectrum of any connected graph of diameter 2 similarities between n as a distance laplacian eigenvalue and the algebraic connectivity are established

replica bounds for optimization problems and diluted spin systems in this paper we generalize to the case of diluted spin models and random combinatorial optimization problems a technique recently introduced by guerra cond mat 0205123 to prove that the replica method generates variational bounds for disordered systems we analyze a family of models that includes the viana bray model the diluted p spin model or random xor sat problem and the random k sat problem showing that the replica cavity method at the various levels of approximation provides systematic schemes to obtain lower bounds of the free energy at all temperatures and of the ground state energy in the case of k sat and xor sat it thus gives upper bounds of the satisfiability threshold our analysis underlines deep connections with the cavity method which are not evident in the long range case

proof for four conjectures about the distance laplacian and distance signless laplacian eigenvalues of a graph abstract the distance laplacian matrix l g of a graph g is defined to be l g diag tr d g where d g denotes the distance matrix of g and diag tr denotes the diagonal matrix of the vertex transmissions in g similarly the distance signless laplacian matrix of g is defined as q g diag tr d g the eigenvalues of l g and q g are called the distance laplacian and distance signless laplacian eigenvalues respectively in this paper four conjectures proposed by m aouchche and p hansen about the largest and the second largest distance laplacian eigenvalues and the second largest distance signless laplacian eigenvalue of a graph are proved

distance spectral radius of uniform hypergraphs abstract we study the effect of three types of graft transformations to increase or decrease the distance spectral radius of connected uniform hypergraphs and we determine the unique k uniform hypertrees with maximum second maximum minimum and second minimum distance spectral radius respectively

distance spectral radius of graphs with r pendent vertices abstract let g n r be the class of all connected graphs of order n with r pendent vertices in this paper we determine the unique graph with minimal distance spectral radius in g n r in addition we determine the unique graph with maximal distance spectral radius in g n r for each r 2 3 n 3 n 2 n 1

extremal cacti of given matching number with respect to the distance spectral radius a cactus is a connected graph in which any two cycles have at most one common vertex the distance spectral radius g of a graph g is the largest eigenvalue of the distance matrix d g recently many researchers proposed the use of g as a molecular structure descriptor of alkanes in this paper we characterize n vertex cyclic cactus with given matching number m which minimizes the distance spectral radius the resulting cactus also minimizes the hosoya index the wiener index and the randic index in the same class of graphs

the effect of graft transformations on distance signless laplacian spectral radius abstract for a connected graph g the distance signless laplacian spectral radius of g is the spectral radius of its distance signless laplacian matrix q g defined as q g t r g d g where t r g is the diagonal matrix of vertex transmissions of g and d g is the distance matrix of g in this paper we study the effect of three types of graft transformations to decrease and or increase the distance signless laplacian spectral radius as applications on one hand we determine the unique graphs with minimum distance signless laplacian spectral radius among non starlike trees among non caterpillar trees and among non starlike non caterpillar trees respectively and determine the unique trees with third and forth minimum distance signless laplacian spectral radius respectively and on the other hand we determine the unique graphs with maximum distance signless laplacian spectral radius among trees with given maximum degree among trees of a perfect matching with maximum degree among non starlike trees among non caterpillar trees among non starlike non caterpillar trees and among unicyclic odd cycle graphs respectively and determine the unique trees with second and third maximum distance signless laplacian spectral radius respectively and the unique connected graph with second maximum distance signless laplacian spectral radius

vector nonlinear schro dinger hierarchies as approximate kadomtsev petviashvili hierarchies abstract the kadomtsev petviashvili kp hierarchy a collection of compatible nonlinear equations each in 2 1 independent variables can be consistently constrained in many different ways to yield hierarchies of equations in 1 1 independent variables in particular the n component vector nonlinear schrodinger vnls hierarchies are contained within the kp hierarchy in this way these hierarchies approximate the kp hierarchy in the limit of large n and this permits the equations of the kp hierarchy to be approximated by nonlinear equations in 1 1 dimensions

sharp bounds on the distance spectral radius and the distance energy of graphs the d eigenvalues 1 2 p of a graph g are the eigenvalues of its distance matrix d and form the d spectrum of g denoted by specd g the greatest d eigenvalue is called the d spectral radius of g denoted by 1 the d energy ed g of the graph g is the sum of the absolute values of its d eigenvalues in this paper we obtain some lower bounds for 1 and characterize those graphs for which these bounds are best possible we also obtain an upperbound for ed g and determine those maximal d energy graphs

on distance matrices and laplacians we consider distance matrices of certain graphs and of points chosen in a rectangular grid formulae for the inverse and the determinant of the distance matrix of a weighted tree are obtained results concerning the inertia and the determinant of the distance matrix of an unweighted unicyclic graph are proved if d is the distance matrix of a tree then we obtain certain results for a perturbation of d 1 as an example it is shown that if l is the laplacian matrix of an arbitrary connected graph then d 1 l 1 is an entrywise positive matrix we consider the distance matrix of a subset of a rectangular grid of points in the plane if we choose m k 1 points not containing a closed path in an m k grid then a formula for the determinant of the distance matrix of such points is obtained

distance matrix and laplacian of a tree with attached graphs a tree with attached graphs is a tree together with graphs defined on its partite sets we introduce the notion of incidence matrix laplacian and distance matrix for a tree with attached graphs formulas are obtained for the minors of the incidence matrix and the laplacian and for the inverse and the determinant of the distance matrix the case when the attached graphs themselves are trees is studied more closely several known results including the matrix tree theorem are special cases when the tree is a star the case when the attached graphs are paths is also of interest since it is related to the transportation problem

broken replica symmetry bounds in the mean field spin glass model by using a simple interpolation argument in previous work we have proven the existence of the thermodynamic limit for mean field disordered models including the sherrington kirkpatrick model and the derrida p spin model here we extend this argument in order to compare the limiting free energy with the expression given by the parisi ansatz and including full spontaneous replica symmetry breaking our main result is that the quenched average of the free energy is bounded from below by the value given in the parisi ansatz uniformly in the size of the system moreover the difference between the two expressions is given in the form of a sum rule extending our previous work on the comparison between the true free energy and its replica symmetric sherrington kirkpatrick approximation we give also a variational bound for the infinite volume limit of the ground state energy per site

maximizing the spectral radius of k connected graphs with given diameter abstract the spectral radius of a graph is the largest eigenvalue of its adjacency matrix p hansen and d stevanovic 2008 9 determined the graphs with maximum spectral radius among all connected graphs of order n with diameter d in this paper we generalize this result to k connected graphs of order n with diameter d

on the distance and distance signless laplacian spectral radii of bicyclic graphs abstract the unique graphs with minimum and second minimum distance distance signless laplacian respectively spectral radii are determined among bicyclic graphs with fixed number of vertices

the generalized distance matrix of digraphs abstract let d g and d q g d i a g t r d g be the distance matrix and distance signless laplacian matrix of a simple strongly connected digraph g respectively where d i a g t r diag d 1 d 2 d n be the diagonal matrix with vertex transmissions of the digraph g to track the gradual change of d g into d q g in this paper we propose to study the convex combinations of d g and d i a g t r defined by d g d i a g t r 1 d g 0 1 this study reduces to merging the distance spectral and distance signless laplacian spectral theories the eigenvalue with the largest modulus of d g is called the d spectral radius of g denoted by g we determine the digraph which attains the maximum or minimum d spectral radius among all strongly connected digraphs moreover we also determine the digraphs which attain the minimum d spectral radius among all strongly connected digraphs with given parameters such as dichromatic number vertex connectivity or arc connectivity

on distance spectral radius of graphs abstract we determine the unique tree of given domination number with minimum distance spectral radius and the unique connected graph tree of given domination number with maximum distance spectral radius we also determine the unique tree of given bipartition size with minimum distance spectral radius and the unique connected bipartite graph tree of given bipartition size with maximum distance spectral radius finally we determine the unique trees with the second and the third minimum maximum respectively distance spectral radii among the trees on n 6 vertices

connectivity diameter independence number and the distance spectral radius of graphs abstract the distance spectral radius of a graph is the largest eigenvalue of its distance matrix x l zhang 2012 31 determined the n vertex graphs of given diameter with the minimum distance spectral radius in this paper we generalize this result by characterizing the graphs of order n with given connectivity and diameter having minimum distance spectral radius in addition we determine the minimum distance spectral radius of graphs among the n vertex graphs with given connectivity and independence number and characterize the corresponding extremal graph thus determining the minimum distance spectral radius of graphs among n vertex graphs with given connectivity resp independence number

exactly solvable disordered sphere packing model in arbitrary dimensional euclidean spaces we introduce a generalization of the well known random sequential addition rsa process for hard spheres in d dimensional euclidean space mathbb r d we show that all of the n particle correlation functions of this nonequilibrium model in a certain limit called the ghost rsa packing can be obtained analytically for all allowable densities and in any dimension this represents the first exactly solvable disordered sphere packing model in an arbitrary dimension the fact that the maximal density ensuremath phi ensuremath infty 1 2 d of the ghost rsa packing implies that there may be disordered sphere packings in sufficiently high d whose density exceeds minkowski s lower bound for bravais lattices the dominant asymptotic term of which is 1 2 d indeed we report on a conjectural lower bound on the density whose asymptotic behavior is controlled by 2 ensuremath 0 778 65 dots d thus providing the putative exponential improvement on minkowski s 100 year old bound our results suggest that the densest packings in sufficiently high dimensions may be disordered rather than periodic implying the existence of disordered classical ground states for some continuous potentials

distance spectral radius of trees with fixed number of pendent vertices abstract the distance spectral radius g of a graph g is the largest eigenvalue of the distance matrix d g in this paper we characterize the graph with minimum distance spectral radius among trees with fixed number of pendent vertices

sharp bounds on the spectral radius of a nonnegative matrix abstract we give upper and lower bounds for the spectral radius of a nonnegative matrix using its row sums and characterize the equality cases if the matrix is irreducible then we apply these bounds to various matrices associated with a graph including the adjacency matrix the signless laplacian matrix the distance matrix the distance signless laplacian matrix and the reciprocal distance matrix some known results in the literature are generalized and improved

distance spectral radius of trees with given number of segments abstract a segment of a tree is a path subtree whose terminal vertices are branching or pendant vertices we determine the structure of trees that maximize the distance spectral radius among trees with given order and number of segments and among trees with given order and number of vertices of degree 2 respectively

gibbs states and the set of solutions of random constraint satisfaction problems abstract an instance of a random constraint satisfaction problem defines a random subset the set of solutions of a large product space x n the set of assignments we consider two prototypical problem ensembles random k satisfiability and q coloring of random regular graphs and study the uniform measure with support on s as the number of constraints per variable increases this measure first decomposes into an exponential number of pure states clusters and subsequently condensates over the largest such states above the condensation point the mass carried by the n largest states follows a poisson dirichlet process for typical large instances the two transitions are sharp we determine their precise location further we provide a formal definition of each phase transition in terms of different notions of correlation between distinct variables in the problem the degree of correlation naturally affects the performances of many search sampling algorithms empirical evidence suggests that local monte carlo markov chain strategies are effective up to the clustering phase transition and belief propagation up to the condensation point finally refined message passing techniques such as survey propagation may also beat this threshold message passing algorithms phase transitions random graphs

nonlinear waves and solitons in physical systems abstract advances in nonlinear science have been plentiful in recent years in particular interest in nonlinear wave propagation continues to grow stimulated by new applications such as fiber optic communication systems as well as the many classical unresolved issues of fluid dynamics what is arguably the turning point for the modern perspective of nonlinear systems took place at los alamos over 40 years ago with the pioneering numerical simulations of fermi pasta and ulam a decade later this research initiated the next major advance of zabusky and kruskal that motivated the revolution in completely integrable systems with this in mind the conference on nonlinear waves in solitions in physical systems nwsps was organized by the center for nonlinear studies cnls at los alamos national laboratory in may of 1997 to assess the current state of the art in this very active field papers from the conference attendees as well as from researchers unable to attend the conference were collected in this special volume of physica d in this paper the contributions to the conference and to this special issue are reviewed with an emphasis on the many unifying principles that all these works share

on the distance spectral radius of cacti abstract a cactus is a connected graph in which any two cycles have at most one common vertex in this article we determine the unique graph with minimal distance spectral radius in the class of all cacti with n vertices and k cycles also we determine the unique graph with minimal distance spectral radius in the class of all cacti with n vertices and r pendent vertices moreover we determine the class of cacti in which the maximal distance spectral radius among all cacti with n vertices and k cycles is attained

linear operators with self consistent coefficients and rational reductions of kp hierarchy abstract a new general type of reductions of the hierarchy of kadomtsev petviashvili equation is established these reductions are equivalent to the lax equations for pseudo differential operators of the form l 1 1 l 2 where l 1 l 2 are ordinary differential operators with coefficients that are functions of variables t 1 x t 2 it is shown that besides the kp hierarchy they are invariant with respect to the rational symmetries as well

time dependent potentials associated with exceptional orthogonal polynomials we introduce a method for generating rational extensions of time dependent potentials such that the associated schrodinger equation admits solutions in terms of exceptional orthogonal polynomials our method is applicable to position dependent schrodinger equations as well as to their conventional counterparts for constant mass

cavity approach to sphere packing in hamming space in this paper we study the hard sphere packing problem in the hamming space by the cavity method we show that both the replica symmetric and the replica symmetry breaking approximations give maximum rates of packing that are asymptotically the same as the lower bound of gilbert and varshamov consistently with known numerical results the replica symmetric equations also suggest a crystalline solution where for even diameters the spheres are more likely to be found in one of the subspaces even or odd of the hamming space these crystalline packings can be generated by a recursive algorithm which finds maximum packings in an ultra metric space finally we design a message passing algorithm based on the cavity equations to find dense packings of hard spheres known maximum packings are reproduced efficiently i n non trivial ranges of dimensions and number of spheres

the generalized distance matrix abstract let d g and d i a g t r denote the distance matrix and diagonal matrix of the vertex transmissions of a simple connected graph g respectively the distance signless laplacian matrix of g is defined as d q g d i a g t r d g heretofore the spectral properties of d g and d q g have attracted much more attention in the present paper we propose to study the convex combinations d g of d i a g t r and d g defined as d g d i a g t r 1 d g 0 1 this study sheds new light on d g and d q g some spectral properties of d g are given and a few open problems are discussed furthermore we take effort to obtain some upper and lower bounds of spectral radius of d g finally the generalized distance spectra of some graphs obtained by operations are also studied

unitary deformations and complex soliton equations the generalized lax equation l i al lb leaves the spectrum of l l invariant if a and b are self adjoint operators consequently such equations possess many conversed quantities with the help of this scheme complex equations of korteweg de vries type are derived

the inertia and energy of distance matrices of complete k partite graphs abstract for a distance matrix d g its inertia is the triple of integers n d n 0 d n d where n d n 0 d n d denote the number of positive 0 negative eigenvalues of d g respectively the d energy is the sum of the absolute eigenvalues of d g in this paper we first study the inertia of distance matrices of complete k partite graphs then as applications we not only prove a conjecture proposed by caporossi et al 2009 in 5 in a different way from stevanovic et al 2013 11 but also obtain the formula of the d energy of the remaining complete k partite graphs at last we obtain the graphs with the maximum resp minimum d energy among all the complete k partite graphs with n vertices

on the maximal distance spectral radius of graphs without a pendent vertex abstract in this article we determine the unique graph with maximal distance spectral radius in the class of graphs without a pendent vertex

the distance eigenvalues of the complements of unicyclic graphs abstract in this paper we determine the unique graph among complements of unicyclic graphs which maximizes the distance spectral radius we also characterize the unique graph among complements of unicyclic graphs of diameter three which maximizes the least distance eigenvalues

computational synergetics and mathematical innovation it is demonstrated how the computer used in a heuristic mode has greatly augmented our understanding of the mathematics of nonlinear dynamical process examples are given of recent work in soliton mathematics mainly but also nonlinear iterative mappings and low order nonlinear ordinary differential equation systems the role of good graphics in enhancing the discovery and retention of new mathematical properties of equations is illustrated several examples are given of the present frontier of research in two space dimensional nonlinear evolutionary problems

convex and quasiconvex functions on trees and their applications abstract we introduce convex and quasiconvex functions on trees and prove that for a tree the eccentricity transmission and weight functions are strictly quasiconvex it is shown that the perron vector of the distance matrix is strictly convex whereas the perron vector of the distance signless laplacian is quasiconvex for a tree in the class of all trees with a given number of pendent vertices we prove that the distance laplacian and distance signless laplacian spectral radius are both maximized at a dumbbell among all trees with fixed maximum degree we prove that the broom is the unique tree that maximizes the distance laplacian and distance signless laplacian spectral radius we find the unique graph that maximizes the distance spectral radius in the class of all unicyclic graphs of girth g on n vertices also we find the unique graph that maximizes the distance signless laplacian and the distance laplacian spectral radius in the class of all unicyclic graphs on n vertices

modal expansions and completeness relations for some time dependent schro dinger equations abstract with the use of a variant of the method of separation of variables the initial value problem for the time dependent linear schrodinger equation is solved exactly for a large class of potential functions related to multisoliton interactions in the vector nonlinear schrodinger equation completeness of states is proved for absolutely continuous initial data in l1

replica bounds for diluted non poissonian spin systems in this paper we extend replica bounds and free energy subadditivity arguments to diluted spin glass models on graphs with arbitrary non poissonian degree distribution the new difficulties specific of this case are overcome introducing an interpolation procedure that stresses the relation between interpolation methods and the cavity method as a byproduct we obtain self averaging identities that generalize the ghirlanda guerra ones to the multi overlap case

on the least eigenvalue of cacti among the cacti with n vertices and k cycles we determine a unique cactus whose least eigenvalue is minimal we also explore cacti with n vertices and among them we find a unique cactus whose least eigenvalue is minimal

on the high density behavior of hamming codes with fixed minimum distance we discuss the high density behavior of a system of hard spheres of diameter d on the hypercubic lattice of dimension n in the limit n d d n the problem is relevant for coding theory and the best available bounds state that the maximum density of the system falls in the interval 1 v d exp n being 0 and v d the volume of a sphere of radius d we find a solution of the equations describing the liquid up to an exponentially large value of widetilde rho v d but we show that this solution gives a negative entropy for the liquid phase for widetilde rho rsimn we then conjecture that a phase transition towards a different phase might take place and we discuss possible scenarios for this transition

dispersionful analog of the whitham hierarchy the transition from the zero genus universal whitham hierarchy to its dispersionful counterpart making use only of the lax representations is presented this is an alternative approach to that of takasaki who has recently shown that the dispersionless limit of the charged multicomponent kadomtsev petviashvili kp hierarchy is the whitham hierarchy the advantage of the presented approach is the construction of finite field reductions which are the main concern in this paper the theory is illustrated by several significant examples

statistical mechanics approach to coding theory we propose a method based on cluster expansion to study the optimal code with a given distance between codewords using this approach we find the gilbert varshamov lower bound for the rate of largest code

the inertia and energy of the distance matrix of a connected graph abstract let g be a connected graph and d g be the distance matrix of g suppose that 1 d 2 d n d are the d eigenvalues of g in this paper we show that the distance matrix of a clique tree is non singular moreover we also prove that the distance matrix of a clique tree has exactly one positive d eigenvalue in addition we determine the extremal graphs with maximum and minimum distance energy among all clique trees

on the distance spectral radius of bipartite graphs abstract in this paper we determine the unique graph with minimum distance spectral radius among all connected bipartite graphs of order n with a given matching number moreover we characterize the graphs with minimal distance spectral radius in the class of all connected bipartite graphs with a given vertex connectivity

random k satisfiability problem from an analytic solution to an efficient algorithm we study the problem of satisfiability of randomly chosen clauses each with k boolean variables using the cavity method at zero temperature we find the phase diagram for the k 3 case we show the existence of an intermediate phase in the satisfiable region where the proliferation of metastable states is at the origin of the slowdown of search algorithms the fundamental order parameter introduced in the cavity method which consists of surveys of local magnetic fields in the various possible states of the system can be computed for one given sample these surveys can be used to invent new types of algorithms for solving hard combinatorial optimizations problems one such algorithm is shown here for the k 3 satisfiability problem with very good performances

on the distance spectra of graphs the distance matrix of a graph g is the matrix containing the pairwise distances between vertices the distance eigenvalues of g are the eigenvalues of its distance matrix and they form the distance spectrum of g we determine the distance spectra of double odd graphs and doob graphs completing the determination of distance spectra of distance regular graphs having exactly one positive distance eigenvalue we characterize strongly regular graphs having more positive than negative distance eigenvalues we give examples of graphs with few distinct distance eigenvalues but lacking regularity properties we also determine the determinant and inertia of the distance matrices of lollipop and barbell graphs

on quadratic poisson brackets in this paper we present a method for constructing large families of quadratic poisson brackets on a manifold using more elementary brackets on a different manifold the method is then applied to several examples of completely integrable systems one can recover several known brackets for systems such as the toda lattice or the open discrete kp hierarchy new brackets for a doubly periodic discrete kp hierarchy are also constructed

on the distance and distance laplacian eigenvalues of graphs abstract let g v e be a simple graph with vertex set v g v 1 v 2 v n and edge set e g let d g be the distance matrix of g for a given nonnegative integer k when n is sufficiently large with respect to k we show that n k d 1 thereby solving a problem proposed by lin et al 2014 8 the distance laplacian spectral radius of a connected graph g is the spectral radius of the distance laplacian matrix of g defined as d l g tr g d g where tr g is the diagonal matrix of vertex transmissions of g aouchiche and hansen 2014 3 conjectured that m 1 d l n 2 when g k n and the equality holds if and only if either g k 1 n 1 or g k n 2 n 2 in this paper we confirm the conjecture

on the distance spectral radius of some graphs abstract for a connected graph the distance spectral radius is the largest eigenvalue of its distance matrix in this paper we determine the unique graph with minimum distance spectral radius among all connected graphs of order n with a given diameter moreover we determine the unique graph with maximum distance spectral radius among the catacondensed hexagonal systems with h hexagons

on least distance eigenvalues of trees unicyclic graphs and bicyclic graphs abstract the connected graphs with least distance eigenvalues in 2 383 0 and the trees with least distance eigenvalues in 2 2 0 have been known we determine the trees with least distance eigenvalues in 3 5 2 2 and the unicyclic and bicyclic graphs with least distance eigenvalues in 2 2 2 383 we also determine the trees with the k th largest least distance eigenvalues for every positive integer k up to 28 and the unicyclic and bicyclic graphs with the k th largest least distance eigenvalues respectively for any positive integer k

instability of one step replica symmetry broken phase in satisfiability problems we reconsider the one step replica symmetry breaking 1rsb solutions of two random combinatorial problems k xorsat and k sat we present a general method for establishing the stability of these solutions with respect to further steps of replica symmetry breaking our approach extends the ideas of montanari and ricci tersenghi 2003 eur phys j b 33 339 to more general combinatorial problems it turns out that 1rsb is always unstable at sufficiently small clause density or high energy in particular the recent 1rsb solution to 3 sat is unstable at zero energy for m with m 4 153 on the other hand the sat unsat phase transition seems to be correctly described within 1rsb