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

analyzing lattice networks through substructures abstract analyzing the topology of network structures is an important topic studied from many different aspects of science and mathematics the wiener polarity index number of unordered pairs of vertices at distance 3 from each other is one of the representative descriptors of graph structures it was computed for several lattice networks by chen et al 11 in an effort to understand the properties of these networks the wiener polarity index is a variation of the classic distance based graph invariant the wiener index sum of distances between all pairs of vertices which is known to be closely related to the number of substructures in this paper we examine the numbers of various subgraphs of order 4 for these lattice networks in addition to confirming their symmetric nature comparing the numbers of various substructures leads to insights on other less trivial characteristics of these network structures of common interest

fluid limits for shortest remaining processing time queues we consider a single server queue with renewal arrivals and i i d service times in which the server uses the shortest remaining processing time policy to describe the evolution of this queue we use a measure valued process that keeps track of the residual service times of all buffered jobs we propose a fluid model or formal law of large numbers approximation for this system and under mild assumptions prove the existence and uniqueness of fluid model solutions furthermore we prove a scaling limit theorem that justifies the fluid model as a first order approximation of the stochastic model the state descriptor of the fluid model is a measure valued function whose dynamics are governed by certain inequalities in conjunction with the standard workload equation in particular these dynamics determine the evolution of the left edge infimum of the state descriptor s support which yields conclusions about response times we characterize the evolution of this left edge as an inverse functional of the initial condition arrival rate and service time distribution this characterization reveals the manner in which the growth rate of the left edge depends on the service time distribution by considering varying examples the authors show that the rate can vary from logarithmic to polynomial

on the laplacian coefficients of tricyclic graphs abstract let g d e t i n l g k 0 n 1 k c k n k be the characteristic polynomial of the laplacian matrix of a graph g of order n in this paper we show that among all connected tricyclic graphs of order n the k th coefficient c k is smallest when the graph is b n 7 1 3 3 3 obtained from the complete graph k 4 by adding n 4 pendent vertices attached to the vertex of degree 3 and for some lemmas in c x he h y shan on the laplacian coefficients of bicyclic graphs discrete math 310 2010 3404 3412 we present a new method to prove them

fluid limits for many server systems with reneging under a priority policy a multiclass many server system is considered in which customers are served according to a nonpreemptive priority policy and may renege while waiting to enter service the service and reneging time distributions satisfy mild conditions building on an approach developed by kaspi and ramanan the law of large numbers many server asymptotics are characterized as the unique solution to a set of differential equations in a measure space regarded as fluid model equations in stationarity convergence to the explicitly solved invariant state of the fluid model equations is established an immediate consequence of the results in the case of exponential reneging is the asymptotic optimality of an index policy called the c rule for the problem of minimizing linear queue length and reneging costs a certain skorohod map plays an important role in obtaining both uniqueness of solutions to the fluid model equations and convergence

mixed problem for some semi linear wave equation with a nonhomogeneous condition on considere le probleme aux valeurs limites et initiales suivant u tt u xx u t sign u t 0 x t 0 1 0 t u x 0 t g t u 1 t 0 u x 0 u 0 x u t x 0 u 1 x avec 0 1 avec u 0 h 1 0 1 u 1 dans l 2 0 1 et g t dans l 2 0 t un theoreme d existence globale et d unicite est demontre

existence and asymptotic expansion for a nonlinear wave equation associated with nonlinear boundary conditions consider the initial boundary value problem for the nonlinear wave equation 1 utt x x t ux f u ut f x t 0 x 1 0 t t 0 t ux 0 t p t 1 t ux 1 t k1 u 1 t p1 2u 1 t ut 1 t q1 2ut 1 t u x 0 u0 x ut x 0 u1 x where p1 q1 2 k1 0 are given constants and u0 u1 f f are given functions and the unknown function u x t and the unknown boundary value p t satisfy the following nonlinear integral equation 2 p t g t k0 u 0 t p0 2u 0 t ut 0 t q0 2ut 0 t 0tk t s u 0 s ds where p0 q0 2 k0 0 are given constants and g k are given functions in this paper we consider three main parts in part 1 under the conditions u0 u1 f g k h1 l2 l1 0 t l2 lq0 0 t l1 0 t c0 qt x t 0 0 t l1 0 t l t x t 0 a e x t qt k0 k1 0 p0 q0 p1 q1 2 q0 q0q0 1 the function f supposed to be continuous with respect to two variables and nondecreasing with respect to the second variable and some others we prove that the problem 1 and 2 has a weak solution u p if in addition k w1 1 0 t p0 p1 2 3 and some other conditions then the solution is unique the proof is based on the faedo galerkin method and the weak compact method associated with a monotone operator for the case of q0 q1 2 p0 p1 2 in part 2 we prove that the unique solution u p belongs to l 0 t h2 c0 0 t h1 c1 0 t l2 h1 0 t with ut l 0 t h1 utt l 0 t l2 u 0 u 1 h2 0 t if we assume u0 u1 h2 h1 f c1 r2 and some other conditions finally in part 3 with q0 q1 2 p0 p1 n 1 f cn 1 r2 n 2 we obtain an asymptotic expansion of the solution u p of the problem 1 and 2 up to order n 1 in two small parameters k0 k1

law of large number limits of limited processor sharing queues motivated by applications in computer and communication systems we consider a processor sharing queue where the number of jobs served is not larger than k we propose a measure valued fluid model for this limited processor sharing queue and show that there exists a unique associated fluid model solution in addition we show that this fluid model arises as the limit of a sequence of appropriately scaled processor sharing queues

mathematical model for a shock problem involving a linear viscoelastic bar are given constants problem 1 1 1 7 describes the shock between a solid body and a linear vis coelastic bar resting on a viscoelastic base with linear elastic constraints at the side constraints associated with a viscous frictional resistance in 1 an and trieu studied a special case of problem 1 1 1 2 1 4 1 7 associated with the following homogeneous boundary condition at x 1 u 1 t 0 1 8 with f u

a wave equation associated with mixed nonhomogeneous conditions global existence and asymptotic expansion of solutions abstract the paper deals with the initial boundary value problem for the linear wave equation 1 u t t u x x k u u t f x t 0 x 1 0 t t u x 0 t p t u 1 t 0 u x 0 u 0 x u t x 0 u 1 x where k are given constants and u 0 u 1 f are given functions and the unknown function u x t and the unknown boundary value p t satisfy the following nonlinear integral equation 2 p t g t k 1 u 0 t 2 u 0 t 1 u t 0 t 2 u t 0 t 0 t k t s u 0 s d s where k 1 1 are given constants and g k are given functions in this paper we consider three main parts in part 1 we prove a theorem of global existence and uniqueness of a weak solution u p of problem 1 1 1 5 the proof is based on a galerkin method associated with a priori estimates weak convergence and compactness techniques for the case of 2 part 2 is devoted the study of the asymptotic behavior of the solution u p as 1 0 finally in part 3 we obtain an asymptotic expansion of the solution u p of the problem 1 1 1 5 up to order n 1 2 in four small parameters k k 1 1

on the energy of some circulant graphs we give an explicit construction of circulant graphs of very high energy this construction is based on gauss sums we also show the littlewood conjecture can be used to establish new result for a certain class of circulant graphs

fluid limits of g g 1 g queues under the nonpreemptive earliest deadline first discipline a single server queueing model is considered with customers that have deadlines if a customer s deadline elapses before service is offered the customer abandons the system customers do not abandon while being served when the server becomes available it offers service to the customer having the earliest deadline among those that are in the queue we obtain a fluid limit of the queue length and abandonment processes and for the occupation measure of deadlines in the form of measure valued processes we characterize the limit by means of a skorohod problem in a time varying domain that has an explicit solution the fluid limits also describe a certain process called the frontier that is well known to play a key role in systems operating under this scheduling policy

existence and asymptotic expansion for a viscoelastic problem with a mixed nonhomogeneous condition abstract we study the initial boundary value problem for a nonlinear wave equation given by 1 u t t u x x 0 t k t s u x x s d s k u p 2 u u t q 2 u t f x t 0 x 1 0 t t u x 0 t u 0 t u x 1 t u 1 t g t u x 0 u 0 x u t x 0 u 1 x where 0 p 2 q 2 k are given constants and u 0 u 1 f g k are given functions in this paper we consider three main parts in part 1 we prove a theorem of existence and uniqueness of a weak solution u of problem 1 the proof is based on a faedo galerkin method associated with a priori estimates weak convergence and compactness techniques part 2 is devoted to the study of the asymptotic behavior of the solution u as 0 finally in part 3 we obtain an asymptotic expansion of the solution u of the problem 1 up to order n 1 in three small parameters k

a nonlinear wave equation associated with nonlinear boundary conditions existence and asymptotic expansion of solutions consider the initial boundary value problem for the nonlinear wave equation equation 1 utt uxx k u p 2u ut q 2ut f x t 0 x 1 0 t t ux 0 t p t ux 1 t u 1 t p1 2u 1 t ut 1 t q1 2ut 1 t u x 0 u0 x ut x 0 u1 x where pp p1p1 q1 2q1 2 q 1q 1 kk are given constants and u0u0 u1u1 ff are given functions and the unknown function u x t u x t and the unknown boundary value p t p t satisfy the following nonlinear integral equation equation 2 p t g t k0 u 0 t p0 2u 0 t ut 0 t q0 2ut 0 t 0tk t s u 0 s ds where p0p0 q0 2q0 2 k0k0 are given constants and gg kk are given functions in this paper we consider three main parts in part 1 under the conditions u0 u1 h1 l2 u0 u1 h1 l2 f l2 qt f l2 qt k w1 1 0 t k w1 1 0 t g lq0 0 t 1 1 kk k0 0k0 0 pp p0p0 q0q0 p1p1 q1 2q1 2 q 1q 1 q0 q0q0 1 we prove a theorem of existence and uniqueness of a weak solution u p u p of problem 1 and 2 the proof is based on the faedo galerkin method and the weak compact method associated with a monotone operator for the case of q0 q1 2q0 q1 2 pp qq p0p0 p1p1 2 2 in part 2 we prove that the unique solution u p u p belongs to l 0 t h2 c0 0 t h1 c1 0 t l2 h1 0 t l 0 t h2 c0 0 t h1 c1 0 t l2 h1 0 t with ut l 0 t h1 ut l 0 t h1 utt l 0 t l2 utt l 0 t l2 u 0 u 0 u 1 h2 0 t u 1 h2 0 t if we make the assumption that u0 u1 h2 h1 u0 u1 h2 h1 and some others finally in part 3 we obtain an asymptotic expansion of the solution u p u p of the problem 1 and 2 up to order n 1n 1 in three small parameters kk k0k0

on a shock problem involving a linear viscoelastic bar abstract the paper deals with the initial boundary value problem for the linear wave equation 1 u tt t u xx ku u t f x t 0 x 1 0 t t u 0 t 0 t u x 1 t q t u x 0 u 0 x u t x 0 u 1 x where k are given constants and u 0 u 1 f are given functions the unknown function u x t and the unknown boundary value q t satisfy the following linear integral equation 2 q t k 1 t u 1 t 1 t u t 1 t g t 0 t k t s u 1 s d s where g k k 1 1 are given functions the paper consists of four parts in part 1 we prove a theorem of global existence and uniqueness of a weak solution u q of problem 1 2 the proof is based on a galerkin type approximation associated to various energy estimates type bounds weak convergence and compactness arguments in part 2 it is devoted to the study of the regularity of the solution u q with respect to the functions 1 f g k k 1 in part 3 we prove that the solution u q of this problem is stable with respect to the data k 1 f g k k 1 finally in part 4 we obtain an asymptotic expansion of the solution u q of this problem up to order n 1 in k for k sufficiently small

graphs and matrices with maximal energy abstract given a complex m n matrix a we index its singular values as 1 a 2 a and call the value e a 1 a 2 a the energy of a thereby extending the concept of graph energy introduced by gutman koolen and moulton proved that e g n 2 1 n for any graph g of order n and exhibited an infinite family of graphs with e g v g 2 1 v g we prove that for all sufficiently large n there exists a graph g g n with e g n 3 2 2 n 11 10 this implies a conjecture of koolen and moulton we also characterize all square nonnegative matrices and all graphs with energy close to the maximal one in particular such graphs are quasi random

the energy of a graph abstract the energy e g of a simple graph g is defined to be the sum of the absolute values of the eigen values of g if g is a k regular graph on n vertices then e g k k n 1 n k b 2 and this bound is sharp it is shown that for each 0 there exist infinitely many n for each of which there exists a k regular graph g of order n with k e g b 2 two graphs with the same number of vertices are equienergetic if they have the same energy we show that for any positive integer n 3 there exist two equienergetic graphs of order 4n that are not cospectral

on the laplacian coefficients of unicyclic graphs abstract let g be a graph of order n and let p g k 0 n 1 k c k n k be the characteristic polynomial of its laplacian matrix generalizing an approach of mohar on graph transformations we show that among all connected unicyclic graphs of order n the k th coefficient c k is largest when the graph is a cycle c n and smallest when the graph is the a s n with an additional edge between two of its pendent vertices a relation to the recently established laplacian like energy of a graph is discussed

on the nonlinear wave equation with the mixed nonhomogeneous conditions linear approximation and asymptotic expansion of solutions in this paper we consider the following nonlinear wave equation equation 1 utt x u ux f x t u ux ut 0 x 1 0 t t ux 0 t g t u 1 t 0 u x 0 u 0 x ut x 0 u 1 x where u 0 u 1 ff gg are given functions to problem 1 we associate a linear recursive scheme for which the existence of a local and unique weak solution is proved by applying the faedo galerkin method and the weak compact method in the case of cn 2 r cn 2 r 1 cn 1 r 1 cn 1 r z 0 0 z 0 0 1 z 0 1 z 0 for all z rz r and g c3 r g c3 r f cn 1 0 1 r r3 f cn 1 0 1 r r3 f1 cn 0 1 r r3 f1 cn 0 1 r r3 a weak solution ue1 e2 x t ue1 e2 x t having an asymptotic expansion of order n 1n 1 in two small parameters e1e1 e2e2 is established for the following equation associated to 1 2 3 equation 2 utt x u e1 1 u ux f x t u ux ut e2f1 x t u ux ut

approximating the gi gi 1 gi queue with a nonlinear drift diffusion hazard rate scaling in heavy traffic we study a single server queue operating under the first in first out fifo service discipline in which each customer independently abandons the queue if his service has not begun within a generally distributed amount of time under some mild conditions on the abandonment distribution we identify a limiting heavy traffic regime in which the resulting diffusion approximation for both the offered waiting time process the process that tracks the amount of time an infinitely patient arriving customer would wait for service and the queue length process contain the entire abandonment distribution to use a continuous mapping approach to establish our weak convergence results we additionally develop existence uniqueness and continuity results for nonlinear generalized regulator mappings that are of independent interest we further perform a simulation study to evaluate the quality of the proposed approximations for the steady state mean queue length and the steady state probability of abandonment suggested by the limiting diffusion process

lower bounds of nikiforov s energy over digraphs abstract the energy of a graph g is defined as e g i 1 n i where 1 2 n are the eigenvalues of the adjacency matrix of g this concept was extended by nikiforov 8 to digraphs as n d i 1 n i where d is a digraph with n vertices and singular values 1 n upper bounds of n were found by kharaghani and tayfeh rezaie 4 in this work we find lower bounds of n over the set of digraphs

a linear wave equation with a nonlinear boundary condition of viscoelastic type abstract the paper is devoted to the study of a linear wave equation with a nonlinear boundary condition of viscoelastic type the existence of a weak solution is proved by using the faedo galerkin method the uniqueness regularity and asymptotic expansion of the solution are also discussed

remarks on the energy of regular graphs abstract the energy of a graph is the sum of the absolute values of the eigenvalues of its adjacency matrix this note is about the energy of regular graphs it is shown that graphs that are close to regular can be made regular with a negligible change of the energy also a k regular graph can be extended to a k regular graph of a slightly larger order with almost the same energy as an application it is shown that for every sufficiently large n there exists a regular graph g of order n whose energy g satisfies g 1 2 n 3 2 n 13 10 several infinite families of graphs with maximal or submaximal energy are given and the energy of almost all regular graphs is determined