on trees with perfect matchings a tree is said to have a perfect matching if it has a spanning forest whose components are paths on two vertices only in this paper we develop upper bounds on the algebraic connectivity of such trees and we consider other eigenvalue properties of its laplacian matrix furthermore for trees with perfect matchings we refine a result due to kirkland neumann and shader concerning the connection between the maximal diagonal entry of the group inverse of the laplacian matrix of a general tree and the pendant vertices of the tree and use this refinement to narrow down the set of the pendant vertices of a tree with a perfect matching which can correspond to the maximal diagonal entry in the group inverse of its laplacian matrix

differentiability of multivalued functions on time scales and applications to multivalued dynamic equations abstract in this paper a class of new derivatives of multivalued functions on a time scale called the hukuhara hilger derivative suited to our study of multivalued dynamic equations is developed moreover some corresponding properties of the hukuhara hilger derivative are discussed the proposed approach forms the appropriate environment within which the study of multivalued dynamic inclusions on time scales can be developed a first example of multivalued dynamic equations is included

the nehari manifold and application to a semilinear elliptic system abstract in this paper we study the nehari manifold and its application to the following semilinear elliptic system u u f x u q 2 u x v v g x v q 2 v x u n h x u 2 u v x v n h x u v 2 v x where is a bounded domain in r n with smooth boundary we prove that the system has at least two nontrivial nonnegative solutions when the pair of the parameters belongs to a certain subset of r 2

trees with minimal laplacian coefficients let g be a simple undirected graph with the characteristic polynomial of its laplacian matrix l g p g m email protected k 0 n 1 kc email protected n k it is well known that for trees the laplacian coefficient c n 2 is equal to the wiener index of g while c n 3 is equal to the modified hyper wiener index of the graph in this paper we characterize n vertex trees with given matching number m which simultaneously minimize all laplacian coefficients the extremal tree a n m is a spur obtained from the star graph s n m 1 with n m 1 vertices by attaching a pendant edge to each of certain m 1 non central vertices of s n m 1 in particular a n m minimizes the wiener index the modified hyper wiener index and the recently introduced incidence energy of trees defined as ie g email protected k 0 email protected k where m k are the eigenvalues of signless laplacian matrix q g d g a g we introduced a general r transformation which decreases all laplacian coefficients simultaneously in conclusion we illustrate on examples of wiener index and incidence energy that the opposite problem of simultaneously maximizing all laplacian coefficients has no solution

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

on the spectral radius of trees with given independence number abstract in the paper we will determine the graphs with maximal spectral radius among all the trees with n vertices and independence number for n 2 n 1

almost periodicity of set valued functions and set dynamic equations on time scales in this paper we first propose the concept of the almost periodicity of set valued functions on almost periodic time scales then based on the exponential dichotomy we discuss the existence and uniqueness of almost periodic solutions of set dynamic equations on time scales as an application several examples are presented to exhibit that one can obtain the existence of almost periodic solutions to interval valued dynamic equations or fuzzy dynamic equations and their discrete counterparts in a united means

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

the nehari manifold for systems of nonlinear elliptic equations abstract this paper deals with the existence and multiplicity results of nonlocal positive solutions to the following system p u u p 1 2 u 1 u u 1 v 1 q v v q 2 v 1 u 1 v 1 v together with dirichlet or mixed boundary conditions under some hypotheses on the parameters p p 1 and q more precisely the system considered corresponds to a perturbed eigenvalue equation combined with a second equation having concave and convex nonlinearities the study is based on the extraction of palais smale sequences in the nehari manifold the behaviour of the energy corresponding to these positive solutions with respect to the real parameters and is established

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

stability and instability for dynamic equations on time scales in this paper we examine the stability and instability of the equilibrium solution x 0 to the first order system of dynamic equations x d f t x t t 0 x d r n where t is from a so called time scale t with t 0 tand d is a compact set our methods involve the existence of a positive definite liapunov function v such that its delta derivative v d satisfies certain integral definite or semidefinite sign properties finally we use liapunov functions to develop an invariance principle regarding solutions to the above dynamic equation

on the extremal values of the eccentric distance sum of trees abstract let g v g e g be a simple connected graph the eccentric distance sum of g is defined as d g v v g e g v d g v where e g v is the eccentricity of the vertex v and d g v u v g d u v is the sum of all distances from the vertex v in this paper we first characterize the extremal trees among n vertex conjugated trees trees with a perfect matching having the minimal and second minimal eccentric distance sums then we identify the trees with the minimal and second minimal eccentric distance sums among the n vertex trees with matching number m finally we characterize the extremal tree with the second minimal eccentric distance sum among the n vertex trees of a given diameter consequently we determine the trees with the third and fourth minimal eccentric distance sums among the n vertex trees which is a continuance study as the results in g h yu l h feng a ilic on the eccentric distance sum of trees and unicyclic graphs j math anal appl 375 2011 99 107

the nehari manifold for a semilinear elliptic system involving sign changing weight functions abstract in this paper we study the combined effect of concave and convex nonlinearities on the number of solutions for a semilinear elliptic system e with sign changing weight function with the help of the nehari manifold we prove that the system has at least two nontrivial nonnegative solutions when the pair of the parameters belongs to a certain subset of r 2

multiple positive solutions for semi linear elliptic systems with sign changing weight abstract in this paper we study the multiplicity results of positive solutions for a semi linear elliptic system involving both concave convex and critical growth terms with the help of the nehari manifold and the lusternik schnirelmann category we investigate how the coefficient h x of the critical nonlinearity affects the number of positive solutions of that problem and get a relationship between the number of positive solutions and the topology of the global maximum set of h

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

stability criteria for set dynamic equations on time scales very recently a new theory known as set dynamic equations on time scales has been built in this paper notions of stability for the solutions of set dynamic equations on time scales using lyapunov like functions are considered criteria for the equistability equiasymptotic stability uniform and uniform asymptotic stability are developed

a remark on the existence of multiple solutions to a multiparameter nonlinear elliptic system abstract in this work motivated by t f wu the nehari manifold for a semilinear elliptic system involving sign changing weight function nonlinear anal 68 2008 1733 1745 and using recent ideas from brown and wu k j brown t f wu a semilinear elliptic system involving nonlinear boundary condition and sign changing weight function j math anal appl 337 2008 1326 1336 we prove the existence of nontrivial nonnegative solutions to the nonlinear elliptic system p u m x u p 2 u u 2 u c x u 2 u v x p v m x v p 2 v v 2 v c x u v 2 v x u 0 v x here p denotes the p laplacian operator defined by p z div z p 2 z p 2 r n is a bounded domain with smooth boundary 1 1 2 p p p p n n p if n p p if n p n is the outer normal derivative r 2 0 0 the weight m x is a bounded function with m 0 and c x is a continuous function which changes sign in

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

on superquadratic elliptic systems in this article we study the existence of solutions for the elliptic system updelta u frac partial h partial v left u v x right text in upomega updelta v frac partial h partial v left u v x right text in upomega u 0 v 0 text on partial upomega where upomega is a bounded open subset of mathbb r n with smooth boundary partial upomega and the function h mathbb r 2 times overline upomega to mathbb r is of class c 1

existence of multiple solutions for quasilinear systems via fibering method existence and multiplicity results for a general class of radial quasilinear equations are proved using the fibering method introduced and developed by s i pohozaev

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

existence of entire positive solutions for a critical system of nonlinear elliptic equations abstract in this paper using the fibering method introduced by pohozaev we establish an existence of multiple nontrivial positive solutions for a system of nonlinear elliptic equations in r n with lack of compactness studying the properties of palais smale sequence of the energy functional associated with the system

combined effects of concave and convex nonlinearities in some elliptic problems abstract this paper deals with a class of semilinear elliptic dirichlet boundary value problems where the combined effects of a sublinear and a superlinear term allow us to establish some existence and multiplicity results

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

two lit trees for lit only σ game abstract a configuration of the lit only game on a finite graph is an assignment of one of two states on or off to all vertices of given a configuration a move of the lit only game on allows the player to choose an on vertex s of and change the states of all neighbors of s given any integer k we say that is k lit if for any configuration the number of on vertices can be reduced to at most k by a finite sequence of moves assume that is a tree with a perfect matching we show that is 1 lit and any tree obtained from by adding a new vertex on an edge of is 2 lit

nehari manifold and existence of positive solutions to a class of quasilinear problems abstract in this paper existence and multiplicity results to the following nonlinear elliptic equation p u u q 2 u u p 2 u u 0 in r n together with mixed dirichlet neumann or neumann boundary conditions are established here p u denotes the p laplacian operator 1 q p n p np n p and is a positive real parameter the study is based on the extraction of palais smale sequences in the nehari manifold

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

on an existence result for a class of p q gradient elliptic systems via a fibering method abstract this paper is devoted to the existence of a solution for a p q gradient elliptic system the main tool employed consists of an application of the fibering method introduced by s i pohozaev

a note on the aα spectral radius 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 2017 7 defined the matrix a g as a g d g 1 a g let u and v be two vertices of a connected graph g suppose that u and v are connected by a path w 0 v w 1 w s 1 w s u where d w i 2 for 1 i s 1 let g p s q u v be the graph obtained by attaching the paths p p to u and p q to v let s 0 1 nikiforov and rojo 2018 9 conjectured that g p s q u v g p 1 s q 1 u v if p q 2 in this paper we confirm the conjecture as applications firstly the extremal graph with maximal a spectral radius with fixed order and cut vertices is characterized secondly we characterize the extremal tree which attains the maximal a spectral radius with fixed order and matching number these results generalize some known results

existence results for some nonlinear elliptic system with lack of compactness abstract we are interested in existence of solutions for some typical elliptic systems governed by two pseudo laplacian operators with critical sobolev exponents we show this result by a local minimization for an adapted variational problem a concentration compactness principle allows us to obtain some convergence results

the nehari manifold for indefinite semilinear elliptic systems involving critical exponent abstract in this paper we study the combined effect of concave and convex nonlinearities on the number of solutions for an indefinite semilinear elliptic system e involving critical exponents and sign changing weight functions using nehari manifold the system is proved to have at least two nontrivial nonnegative solutions when the pair of the parameters belongs to a certain subset of r 2

exponential stability for set dynamic equations on time scales abstract a new theory known as set dynamic equations on time scales has been built the criteria for the equistability equiasymptotic stability uniform and uniformly asymptotic stability were developed in hong 2010 1 in this paper we consider the exponential stability exponentially asymptotic stability uniform and uniformly exponentially asymptotic stability for the trivial solution of set dynamic equations on time scales by using lyapunov like functions

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

existence results to elliptic systems with nonstandard growth conditions abstract nonstandard growth conditions in partial differential equations have been the subject of recent developments in elastic mechanics and electrorheological fluid dynamics lecture notes in mathematics vol 1748 2000 c r acad sci paris ser i math 329 1999 393 398 math ussr izv 29 1987 33 66 in this work elliptic systems with nonstandard growth conditions are studied existence and multiplicity results under growth conditions on the reaction terms are established

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

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

the nehari manifold for a semilinear elliptic equation with a sign changing weight function abstract the nehari manifold for the equation u x a x u x b x u x 1u x for x together with dirichlet boundary conditions is investigated exploiting the relationship between the nehari manifold and fibrering maps i e maps of the form t j tu where j is the euler functional associated with the equation we discuss how the nehari manifold changes as changes and show how existence and non existence results for positive solutions of the equation are linked to properties of the manifold

phase spaces and periodic solutions of set functional dynamic equations with infinite delay abstract very recently a new theory known as set dynamic equations on time scales has been built in this paper a phase space is built for set functional dynamic equations with infinite delay on time scales and sufficient criteria are established for the existence of periodic solutions of such equations which generalize and incorporate as special cases some known results for set differential equations and for set difference equations when the time scale is the real number set or the integer set respectively moreover for differential inclusions and difference inclusions if the variable under consideration is a single valued mapping our results show that one can unify the study of some continuous or discrete problems in the sense of set dynamic equations on general time scales

critical growth fractional elliptic systems with exponential nonlinearity abstract we study the existence of positive solutions for the system of fractional elliptic equations of the type 1 2 u p p q f x u p 2 u v q h 1 u v e u 2 v 2 in 1 1 1 2 v q p q f x u p v q 2 v h 2 u v e u 2 v 2 in 1 1 u v 0 in 1 1 u v 0 in r 1 1 where 1 p q 2 h 1 u v 2 u 2 u 2 u v h 2 u v 2 v 2 u v 2 v and 2 here 1 2 is the fractional laplacian operator we show the existence of multiple solutions for suitable range of by analyzing the fibering maps and the corresponding nehari manifold we also study the existence of positive solutions for a superlinear system with critical growth exponential nonlinearity

the nehari manifold for a class of concave convex elliptic systems involving the p laplacian and nonlinear boundary condition abstract the existence and multiplicity of weak solutions is established for a class of concave convex elliptic systems of the form p u m x u p 2 u a x u 2 u x p v m x v p 2 v b x v 2 v x u p 2 u n u 2 u v v p 2 v n u v 2 v x here p denotes the p laplacian operator defined by p z div z p 2 z p 2 r n is a bounded domain with smooth boundary 1 1 2 p p p p n n p if n p p if n p n is the outer normal derivative r 2 0 0 the weight m x is a positive bounded function and a x b x c are functions which change sign in our technical approach is based on the nehari manifold which is similar to the fibering method of drabek and pohozaev 1997 29 together with the recent idea from brown and wu 2008 10

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

on the index of tricyclic graphs with perfect matchings abstract let t 2 k be the set of all tricyclic graphs on 2 k k 2 vertices with perfect matchings in this paper we discuss some properties of the connected graphs with perfect matchings and then determine graphs with the largest index in t 2 k

on the exponential stability of dynamic equations on time scales in this paper we deal with some theorems on the exponential stability of trivial solution of time varying non regressive dynamic equation on time scales with bounded graininess in particular well known perron s theorem is generalized on time scales under rather restrictive condition that is integral boundedness of coefficient operators we obtain a characterization of the uniformly exponential stability

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

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

stability of set differential equations and applications abstract the paper introduces some results on stability of sheaf solutions exponential stability of solutions and sheaf solutions of set differential equations and their applications to set control differential equations

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

dynamics and stability of impulsive hybrid setvalued integro differential equations with delay abstract we study the dynamics and stability theory for impulsive hybrid set integro differential equations with delay sufficient conditions for the stability of the null solution of impulsive hybrid set integro differential equations with delay are presented

semilinear elliptic problems with mixed dirichlet neumann boundary conditions this work deals with the analysis of problems u uq urin u 0in b u 0on where b u means mixed boundary conditions we prove results about existence multiplicity and a priori estimates

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

multiple solutions for a class of biharmonic elliptic systems with sobolev critical exponent abstract in this paper we study the existence and multiplicity of nontrivial solutions for a class of biharmonic elliptic systems with sobolev critical exponent in a bounded domain by using the variational method and the nehari manifold we obtain the existence and multiplicity results of nontrivial solutions for the systems