the algebraic entropy of classical mechanics we describe the lie algebra of classical mechanics modelled on the lie algebra generated by kinetic and potential energy of a simple mechanical system with respect to the canonical poisson bracket it is a polynomially graded lie algebra a class we introduce we describe these lie algebras give an algorithm to calculate the dimensions cn of the homogeneous subspaces of the lie algebra of classical mechanics and determine the value of its entropy limn cn1 n it is 1 825 423 774 201 08 a fundamental constant associated with classical mechanics

universal equivalence of partially commutative metabelian lie algebras abstract in this paper we find a criterion for universal equivalence of partially commutative lie algebras whose defining graphs are trees besides we obtain bases for partially commutative metabelian lie algebras

the distance spectrum of complements of trees abstract let g be a connected graph of order n and d g be its distance matrix in this paper we characterize the unique graphs whose distance spectral radius attains the maximum and minimum among all complements of trees furthermore we determine the unique graphs whose least distance eigenvalues attains the maximum and minimum among all complements of trees

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

the least eigenvalue of graphs whose complements have only two pendent vertices abstract let g be a simple graph and a g be the adjacency matrix of g the eigenvalues of a g are referred to as the eigenvalues of g in this paper we characterize the graphs with the minimal least eigenvalue among all graphs whose complements are connected and have only two pendent vertices

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

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

the graphs with the least distance eigenvalue at least 1 172 let g be a connected graph with order n and d g be the distance matrix of g suppose that 1 d g n d g are the distance spectra of g in this paper we characterize the graphs with n d g 1 172 1 1 1 2 where is the smallest root of x3 x2 3x 1 0 and 1 172 1 1 2 furthermore we show that the graphs with n 1 172 are determined by their d spectrum

a qualitative explanation of the origin of torsional instability in suspension bridges abstract we consider a mathematical model for the study of the dynamical behavior of suspension bridges we show that internal resonances which depend on the bridge structure only are the origin of torsional instability we obtain both theoretical and numerical estimates of the thresholds of instability our method is based on a finite dimensional projection of the phase space which reduces the stability analysis of the model to the stability of suitable hill equations this gives an answer to a long standing question about the origin of torsional instability in suspension bridges

the least eigenvalue of a graph with cut vertices in this paper we characterize the unique graph whose least eigenvalue attains the minimum among all connected graphs of fixed order and given number of cut vertices and then obtain a lower bound for the least eigenvalue of a connected graph in terms of the number of cut vertices during the discussion we also get some results for the spectral radius of a connected bipartite graph and its upper bound

the least eigenvalue of the complements of trees let tnc be the set of the complements of trees of order n in this paper we characterize the unique graph whose least eigenvalue attains the minimum among all graphs in tnc

the stability of the equilibrium of a nonlinear planar system and application to the relativistic oscillator article history in this paper we present a sufficient condition for the stability of the equilibrium of a nonlinear planar system the proof is based on the computation of the corresponding birkhoff normal forms the result does not involve small parameters applications to the relativistic oscillator are also given

free partially commutative structures abstract in this paper we study algebraic structures defined by a presentation of the form a ab ba for a b i we show that these relators are the only one that can be interpreted as both monoid and lie relators we also study the free partially commutative monoids lie algebras and groups in all of these cases we show how to obtain decomposition results for these structures into the corresponding free structures

on the support of graph lie algebras we deal with the support of graph lie algebras and we characterize the set of independence alphabets for which the noncommutative result can naturally be extended we also provide an algorithm which decides if an independence alphabet fulfills the preceding condition

blow up oscillating solutions to some nonlinear fourth order differential equations abstract we give strong theoretical and numerical evidence that solutions to some nonlinear fourth order ordinary differential equations blow up in finite time with infinitely many wild oscillations we exhibit an explicit example where this phenomenon occurs we discuss possible applications to biharmonic partial differential equations and to the suspension bridges model in particular we give a possible new explanation of the collapse of bridges

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

commutativity and spectra of hermitian matrices abstract if two hermitian matrices commute then the eigenvalues of their sum are just the sums of the eigenvalues of the two matrices in a suitable order examples show that the converse is not true in general in this paper partial converses are obtained the technique involves a characterization of the equality cases for weyl s inequalities moreover a new proof on the commutativity of two hermitian matrices with property l and analogous results for the product of two positive definite hermitian matrices are included

on the partially commutative shuffle product abstract we give recursive formulas that allow us to compute the partially commutative shuffle product of two partially commutative words

wide oscillation finite time blow up for solutions to nonlinear fourth order differential equations we give sufficient conditions for local solutions to some fourth order semilinear ordinary differential equations to blow up in finite time with wide oscillations a phenomenon not visible for lower order equations the result is then applied to several classes of semilinear partial differential equations in order to characterize the blow up of solutions including in particular its applications to a suspension bridge model we also give numerical results which describe this oscillating blow up and allow us to suggest several open problems and to formulate some related conjectures

on least eigenvalues and least eigenvectors of real symmetric matrices and graphs abstract we give an upper bound for the least eigenvalue of a principal submatrix of a real symmetric matrix with zero diagonal from which we establish an upper bound for the least eigenvalue of a graph when some vertices are removed using the components of the least eigenvector s we give lower and upper bounds for the least eigenvalue of a graph when some edges are removed we also establish bounds for the components of the least eigenvector s of a real symmetric matrix and a graph

the complement of the path is determined by its spectrum abstract it is proved that a graph whose 0 1 adjacency matrix has the spectrum of p n the complement of the path on n vertices must be p n

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

sharp lower bounds of the least eigenvalue of planar graphs let g be a simple graph with n 3 vertices and orientable genus g and non orientable genus h we define the euler characteristic g of a graph g by g max 2 2g 2 h let g be the least eigenvalue of the adjacency matrix a of g in this paper we obtain the following lower bounds of g g 2 n g in particular if g is the planar graph then g 2n 4 the equality holds if and only if g k2 n 2 further we have same result of series parallel graph

on the spectral radius of 0 1 matrices we determine the maximum spectral radius for 0 1 matrices with k2 andk2 1 1 s respectively and for symmetric 0 1 matrices with zero trace and e k21 s graphs with e edges in all cases equality is characterized

lower bounds on the spectra of symmetric matrices with nonnegative entries if we normalize a symmetric n n matrix with nonnegative entriesso that its largest entry is 1 then its spectrum is bounded below by n2 the lower bound is achieved in all even dimensions for and only for adjacency matrices of complete bipartite graphs with equal parts

on the second largest aα eigenvalues of graphs abstract let g be a graph with adjacency matrix a g and the degree diagonal matrix d g for any real 0 1 nikiforov 2017 10 defined the matrix a g as a g d g 1 a g denote its eigenvalues by 1 a g 2 a g n a g in this paper we give a lower bound of 2 a g for 0 1 as an application we show that the star is the unique graph which minimizes the second largest a eigenvalues among connected graphs for 1 2 1 in particular the graphs with k a g n 2 for k 2 are determined furthermore we determine the connected graphs with 2 a g 1 are firefly graphs moreover the firefly graphs are shown to be determined by their a spectrum for 1 2 1

initial boundary value problem for the nonlinear string beam system this paper deals with a nonlinear string beam system describing the torsional vertical oscillations of a suspension bridge we consider the initial boundary value problem and study the existence and uniqueness question we assume time independent right hand sides but allow quite general nonlinear terms using the faedo galerkin method we prove the existence of a unique solution on an arbitrary large time interval

qualitative behavior of global solutions to some nonlinear fourth order differential equations abstract we study global solutions to a fourth order semilinear ordinary differential equation we determine sufficient conditions on the nonlinearity that ensure global continuation of the solutions furthermore we discuss their qualitative behaviors such as oscillations and boundedness

eigenvalue interlacing for certain classes of matrices with real principal minors abstract two common properties of z matrices and hermitian matrices are considered 1 the eigenvalue interlacing property i e the two smallest real eigenvalues of a matrix are interlaced by the smallest real eigenvalue of any principal submatrix of order one less 2 the positive glp property i e if a matrix has a positive sequence of generalized leading principal minors then all the principal minors of the matrix are positive the relationship between these properties as well as related properties is examined in general

parabolic and quasiparabolic subgroups of free partially commutative groups let be a finite graph and g be the corresponding free partially commutative group in this paper we study subgroups generated by vertices of the graph which we call canonical parabolic subgroups a natural extension of the definition leads to canonical quasiparabolic subgroups it is shown that the centralisers of subsets of g are the conjugates of canonical quasiparabolic centralisers satisfying certain graph theoretic conditions

on the aα spectra of graphs abstract let g be a graph with adjacency matrix a g and let d g be the diagonal matrix of the degrees of g for any real 0 1 nikiforov 8 defined the matrix a g as a g d g 1 a g in this paper we give some results on the eigenvalues of a g for 1 2 in particular we characterize the graphs with k a g n 1 for 2 k n moreover we show that n a g 2 1 if g contains no isolated vertices

new results on the dα matrix of connected graphs abstract let g be a simple undirected connected graph let d g be the distance matrix of g and let tr g be the diagonal matrix of the vertex transmissions in g let 0 1 in s y cui et al 2019 7 the matrix d g tr g 1 d g is introduced and several properties are obtained in this paper new properties on the d matrix are derived including inequalities that involve the largest vertex transmission and the spectral radii of the distance matrix distance signless laplacian matrix and d matrix the necessary and sufficient condition for the equality in each of the inequalities is given moreover some results on the d matrix of a graph with independent sets of vertices sharing the same set of neighbors including the case of a complete multipartite graph are obtained finally the spectrum of d g is determined when g is the h join of regular graphs

on the least eigenvalue of aα matrix of graphs abstract let g be a graph of order n with m edges and chromatic number let a g be the adjacency matrix and d g be the diagonal matrix of vertex degrees of g nikiforov defined the matrix a g as a g d g 1 a g where 0 1 then a 1 2 g 1 2 d g a g 1 2 q g where q g is the signless laplacian matrix of g let q n g and n a be the least eigenvalue of q g and a g respectively lima et al 2011 8 proposed some open problems on q n g two of which are as follows 1 to characterize the graphs for which q n g 2 m n 1 2 to characterize the graphs for which q n g 1 1 1 2 m n in this paper we present an upper bound on n a in terms of n m and 1 2 1 and characterize the extremal graphs as an application we completely solve problem 1 moreover we examine the more generalized result of problem 2 on a g when 1 we obtain some necessary conditions for n a 1 2 m 1 n and as a corollary for the equality in problem 2

the shuffle algebra and its derivations abstract the shuffle algebra on a partially commutative alphabet and its derivations are studied the derivation of the shuffle algebra is characterized in terms of admissible mappings the module of all derivations of the shuffle algebra is proved to be isomorphic to the free partially commutative power series lie algebra with the coefficient ring extended the orthogonality between the lie elements and the proper shuffles is established

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

blow up profile for solutions of a fourth order nonlinear equation abstract it is well known that the nontrivial solutions of the equation u r u r f u r 0 blow up in finite time under suitable hypotheses on the initial data and f these solutions blow up with large oscillations knowledge of the blow up profile of these solutions is of great importance for instance in studying the dynamics of suspension bridges the equation is also commonly referred to as extended fisher kolmogorov equation or swift hohenberg equation in this paper we provide details of the blow up profile the key idea is to relate this blow up profile to the existence of periodic solutions for an auxiliary equation

minimizing the least eigenvalues of unicyclic graphs with application to spectral spread the spread of a graph is defined to be the difference between the largest eigenvalue and the least eigenvalue of the adjacency matrix of the graph let u k denote the set of connected unicyclic graphs of order n and girth k and let un denote the set of connected unicyclic graphs of order n in this paper we determine the unique graph with minimum least eigenvalue respectively the unique graph with maximum spread among all graphs in u k we finally characterize the unique graph with minimum least eigenvalue respectively the unique graph with maximum spread among all graphs in un

the least eigenvalue of graphs with given connectivity abstract let g be a simple graph and a g be the adjacency matrix of g the eigenvalues of g are those of a g in this paper we characterize the graphs with the minimal least eigenvalue among all graphs of fixed order with given vertex connectivity or edge connectivity

further developments on brouwer s conjecture for the sum of laplacian eigenvalues of graphs abstract let g be a simple graph with order n and size m and having laplacian eigenvalues 1 2 n 1 n 0 and let s k g i 1 k i be the sum of k largest laplacian eigenvalues of g brouwer conjectured that s k g m k 1 2 for all k 1 2 n we obtain upper bounds for s k g in terms of the clique number the order n and integers p 0 r 1 s 1 s 2 2 associated to the structure of the graph g we discuss brouwer s conjecture for two large families of graphs the first family of graphs is obtained from a clique of size by identifying each of its vertices to a vertex of an arbitrary c cyclic graph and the second family is composed of the graphs in which the removal of the edges of the maximal complete bipartite subgraph gives a graph each of whose non trivial components is a c cyclic graph we show among these two large families of graphs the brouwer s conjecture holds for various subfamilies of graphs depending upon the value of c the order of the c cyclic graphs the clique number of the graph the order of the maximal complete bipartite subgraph and the number of the c cyclic components of the graph

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

graphs with three distinct eigenvalues and largest eigenvalue less than 8 abstract in this paper we consider graphs with three distinct eigenvalues and we characterize those with the largest eigenvalue less than 8 we also prove a simple result which gives an upper bound on the number of vertices of graphs with a given number of distinct eigenvalues in terms of the largest eigenvalue

nordhaus gaddum type inequalities of the second aα eigenvalue of a graph abstract let g be a graph with adjacency matrix a g and the degree diagonal matrix d g for any real 0 1 the matrix a g of a graph is defined as a g d g 1 a g denote its eigenvalues by 1 a g 2 a g n a g in this paper for 1 2 1 the nordhaus gaddum type bound for the second largest a eigenvalue of a graph is considered we show that 2 a g 2 a g n 1 and the extremal graphs for which the equality holds are k n and s n if g k n s n k n e then 2 a g 2 a g n 1 moreover we give two upper nordhaus gaddum type bounds for the second largest a eigenvalue of a graph

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

an instability result in the theory of suspension bridges abstract we consider a second order system of two odes which arises as a single mode galerkin projection of the so called fish bone berchio and gazzola 2015 model of suspension bridges the two unknowns represent flexural and torsional modes of vibration of the deck of the bridge the elastic response of the cables is supposed to be asymptotically linear under traction and asymptotically constant when compressed a generalization of the slackening regime we establish a condition depending on a set of 3 parameters under which the flexural motions are unstable provided the energy is sufficiently large

signless laplacian eigenvalue problems of nordhaus gaddum type abstract let g be a graph of order n and let q 1 g q 2 g q n g denote the signless laplacian eigenvalues of g ashraf and tayfeh rezaie 2014 3 showed that q 1 g q 1 g 3 n 4 with equality holding if and only if g or g is the star k 1 n 1 in this paper we prove that q 2 g q 2 g 2 n 4 where the equality holds if and only if g or g is k 2 p 4 or c 4 also we discuss the following problem for n 6 does q 2 g q 2 g 2 n 5 always hold we provide positive answers to this problem for the graphs with disconnected complements and the bipartite graphs and determine the graphs attaining the bound moreover we show that q 2 g q 2 g n 2 and the extremal graphs are also characterized

on the distance spectrum of graphs 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 characterize all connected graphs with n d 2 furthermore we characterize all connected graphs of diameter 2 with exactly three d eigenvalues when 1 d is not an integer we also conjecture that the complete k partite graph is determined by its d spectrum

on the instability tongues of the hill equation coupled with a conservative nonlinear oscillator abstract we study the asymptotics for the lengths l n q of the instability tongues of hill equations that arise as iso energetic linearization of two coupled oscillators around a single mode periodic orbit we show that for small energies i e q 0 the instability tongues have the same behavior that occurs in the case of the mathieu equation l n q o q n the result follows from a theorem which fully characterizes the class of hill equations with the same asymptotic behavior in addition in some significant cases we characterize the shape of the instability tongues for small energies motivation of the paper stems from recent mathematical works on the theory of suspension bridges

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

bicyclic graphs for which the least eigenvalue is minimum abstract the spread of a graph is defined to be the difference between the greatest eigenvalue and the least eigenvalue of the adjacency matrix of the graph in this paper we determine the unique graph with minimum least eigenvalue among all connected bicyclic graphs of order n also we determine the unique graph with maximum spread in this class for each n 28

on the aα spectra of trees abstract let g be a graph with adjacency matrix a g and let d g be the diagonal matrix of the degrees of g for every real 0 1 define the matrix a g as a g d g 1 a g this paper gives several results about the a matrices of trees in particular it is shown that if t is a tree of maximal degree then the spectral radius of a t satisfies the tight inequality a t 2 1 1 which implies previous bounds of godsil lovasz and stevanovic the proof is deduced from some new results about the a matrices of bethe trees and generalized bethe trees in addition several bounds on the spectral radius of a of general graphs are proved implying tight bounds for paths and bethe trees

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

the vertex edge independence number vertex edge cover number and the least eigenvalue of a graph in this paper we characterize the unique graph whose least eigenvalue attains the minimum among all graphs of a fixed order and a given vertex edge independence number or vertex edge cover number and get some bounds for the vertex edge independence number vertex edge cover number of a graph in terms of the least eigenvalue of the graph

on the least distance eigenvalue of a graph abstract denote by d g d i j n n the distance matrix of a connected graph g with n vertices where d i j is equal to the distance between vertices v i and v j in g the least eigenvalue of d g is called the least distance eigenvalue of g denoted by n in this paper we determine all the graphs with n 2 383 0

the free partially commutative lie algebra bases and ranks abstract in this paper we study the free partially commutative lie k algebra l a defined by a commutation relation on an alphabet a its behavior is very similar to that of the free lie algebra indeed we obtain in particular a partially commutative version of lazard s elimination process which allows us to prove that the k module l a is free and to construct explicitly k bases for it we show also how the classical witt s calculus can be extended to l a

on the second largest eigenvalue of the signless laplacian abstract let g be a graph of order n and let q 1 g q n g be the eigenvalues of the q matrix of g also known as the signless laplacian of g we give a necessary and sufficient condition for the equality q k g n 2 where 1 k n in particular this result solves an open problem raised by wang belardo huang and borovicanin we also show that q 2 g g and determine all graphs for which equality holds

eigenvalue inequalities and equalities abstract we consider cases of equality in three basic inequalities for eigenvalues of hermitian matrices cauchy s interlacing inequalities for principal submatrices weyl s inequalities for sums and the residual theorem several applications generalize and sharpen known results for eigenvalues of irreducible tridiagonal hermitian matrices

the algebraic connectivity of graphs with given circumference abstract the algebraic connectivity of a graph is the second smallest eigenvalue of its laplacian matrix algebraic connectivity is closely related to the traditional vertex edge connectivity and it plays an important role in the design of various networks in this paper we determine the graph which has the minimum algebraic connectivity among all connected graphs with given circumference

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

characterization of extremal graphs from distance signless laplacian eigenvalues abstract let g v e be a connected graph with vertex set v g v 1 v 2 v n and edge set e g the transmission tr v i of vertex v i is defined to be the sum of distances from v i to all other vertices let tr g be the n n diagonal matrix with its i i entry equal to tr g v i the distance signless laplacian is defined as d q g tr g d g where d g is the distance matrix of g let 1 g 2 g n g denote the eigenvalues of distance signless laplacian matrix of g in this paper we first characterize all graphs with n g n 2 secondly we characterize all graphs with 2 g n 2 n when n 11 furthermore we give the lower bound on 2 g with independence number and the extremal graph is also characterized

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