the aα spectral radius of trees and unicyclic graphs with given degree sequence abstract for any real 0 1 a g d g 1 a g is the a matrix of a graph g where a g is the adjacency matrix of g and d g is the diagonal matrix of the degrees of g this paper presents some extremal results about the spectral radius 1 a g of a g that generalize previous results about 1 a0 g and 1 a 1 2 g in this paper we study the behavior of the a spectral radius under some graph transformations for 0 1 as applications we show that the greedy tree has the maximum a spectral radius in g d when d is a tree degree sequence firstly furthermore we determine that the greedy unicyclic graph has the largest a spectral radius in g d when d is a unicyclic graphic sequence where g d g g is connected with d as its degree sequence

proof of a conjecture on the spectral radius of c4 free graphs abstract let g be a graph with n vertices and be the spectral radius of its adjacency matrix recently nikiforov showed that if g has no 4 cycle then 2 n 1 0 with equality if and only if g is the friendship graph however this bound is not attainable when n is even he conjectured that if g is a c 4 free graph with even number of vertices then 3 2 n 1 1 0 with equality if and only if g is a star of order n with n 2 1 disjoint additional edges we prove the conjecture in this paper

on the spectral radius of unicyclic graphs with prescribed degree sequence we consider the set of unicyclic graphs with prescribed degree sequence in this set we determine the unique graph with the largest spectral radius or index with respect to the adjacency matrix in addition we give a conjecture about the unique graph with the largest index in the set of connected graphs with prescribed degree sequence

the hirsch conjecture for dual transportation polyhedra an algorithm is given that joins any pair of extreme points of a dual transportation polyhedron by a path of at most m 1 n 1 extreme edges

a relaxation column signature method for assignment problems abstract a recent computational study 7 on algorithms for assignment problems revealed that the so called modified hung rom algorithm mhra performs very well in practice the mhra is a row signature method and its effectiveness comes mainly from its capability to reduce the level by more than one unit per cycle we present an o n 3 column signature method which has the above mentioned capability the algorithm solves a sequence of relaxed transportation problems and generates strong dual feasible bases it is shown that the average number of iterations is bounded by n log n a result on balinski s strong dual feasible bases is also presented

maximizing the spectral radius of graphs with fixed minimum degree and edge connectivity abstract the spectral radius g of a graph g is the largest eigenvalue of the adjacency matrix a g suppose a graph g 0 maximizes the spectral radius over the class of graphs of order n with fixed minimum degree and edge connectivity in this paper we mainly show that g 0 b n where b n is obtained by adding edges between k 1 and k n 1 a property of the adjacency matrix of g 0 is also obtained moreover graphs that maximize g over the class of graphs with minimum degree and edge connectivity for 0 1 2 3 are completely determined

a domain decomposition method for the polar factorization of vector valued mappings we prove the convergence of an overlapping domain decomposition method for the polar factorization of vector valued mappings we introduce a consistent discretization of this problem under the form of an assignment problem the domain decomposition method applied to the discrete problem does not necessarily produce the wanted discrete solution however we prove that the sequence of outputs it produces when we refine the discretization tends to the solution of the continuous problem

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

further results on the least eigenvalue of connected graphs in this paper we identify within connected graphs of order n and size n k with 0 k 4 and n k 5 the graphs whose least eigenvalue is minimal it is also observed that the same graphs have the largest spectral spread if n is large enough

a combinatorial algorithm for the euler equations of incompressible flows abstract a lagrangian time discretization is introduced for the euler equations of inviscid incompressible flows using the least square projection onto the manifold of volume preserving diffeomorphisms a further discretization is performed first by splitting the physical domain into n blocks of equal volume then by approximating each volume preserving mapping involved in the time discretization by a permutation of the n blocks finally a combinatorial algorithm is obtained where the classical assignment problem has to be solved at each time step

the majorization theorem of connected graphs abstract let d 1 d 2 d n and d 1 d 2 d n be two non increasing degree sequences we say is majorizated by denoted by if and only if i 1 n d i i 1 n d i and i 1 j d i i 1 j d i for all j 1 2 n if the degree of vertex v is resp not equal to 1 then we call v a pendant resp non pendant vertex of g we use c to denote the class of connected graphs with degree sequence suppose and are two non increasing c cyclic degree sequences let g and g be the graphs with greatest spectral radii in c and c respectively in this paper we shall prove that if g and g have the same number of pendant vertices and the degrees of all non pendant vertices of g are greater than c then g g

a parallel shortest augmenting path algorithm for the assignment problem abstract we describe a parallel version of the shortest augmenting path algorithm for the assignment problem while generating the initial dual solution and partial assignment in parallel does not require substantive changes in the sequential algorithm using several augmenting paths in parallel does require a new dual variable recalculation method the parallel algorithm was tested on a 14 processor butterfly plus computer on problems with up to 900 million variables the speedup obtained increases with problem size the algorithm was also embedded into a parallel branch and bound procedure for the traveling salesman problem on a directed graph which was test on the butterfly plus on problems involving up to 7 500 cities to our knowledge these are the largest assignment problems and traveling salesman problems solved so far

a new strongly polynomial dual network simplex algorithm this paper presents a new dual network simplex algorithm for the minimum cost network flow problem the algorithm works directly on the original capacitated network and runs in o mn m n logn logn time for the network withn nodes andm arcs this complexity is better than the complexity of orlin plotkin and tardos 1993 dual network simplex algorithm by a factor ofm n

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

transportation problems which can be solved by the use of hirsch paths for the dual problems balinski uses his signature method for the proof of the hirsch conjecture for dual transportation polyhedra to obtain an efficient algorithm for the assignment problem we will show how to extend this method to other primal transportation problems including transportation problems with unit demands we then prove that balinski s assignment algorithm is equivalent cycle by cycle to that of hung and rom we demonstrate that under some assumptions for our probability model a modification of the latter algorithm has an average complexity of o n2logn and present some computational results confirming this we also present results that indicate that this modification compares favorably with balinski s algorithm and other codes

technical note a polynomial simplex method for the assignment problem we present a polynomial primal simplex algorithm for the assignment problem for n n assignment problem with integer cost coefficients the algorithm generates at most n3 ln bases prior to reach the optimal basis where is the difference in objective value between an initial extreme point and the optimal extreme point

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

the signless laplacian spectral radius of tricyclic graphs and trees with k pendant vertices abstract in this paper we consider the following problem of all tricyclic graphs or trees of order n with k pendant vertices n k fixed which achieves the maximal signless laplacian spectral radius we determine the graph with the largest signless laplacian spectral radius among all tricyclic graphs with n vertices and k pendant vertices then we show that the maximal signless laplacian spectral radius among all trees of order n with k pendant vertices 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 to one end vertex of one path we also discuss the signless laplacian spectral radius of t n k and give some results

weak existence for the semigeostrophic equations formulated as a coupled monge ampere transport problem hoskins s semigeostrophic equations are reformulated as a coupled monge ampere transport problem b j hoskins quart j royal met soc 97 1971 pp 139 153 existence of global weak solutions is obtained for this formulation

on ordinary and signless laplacian spectral radius of graphs with fixed number of branch vertices the unique graphs with maximum spectral radius and signless laplacian spectral radius are determined among trees unicyclic graphs and bicyclic graphs respectively with fixed numbers of vertices and branch vertices vertices of degree at least 3 using a unified approach

an infeasible exterior point simplex algorithm for assignment problems the so called modified hung rom algorithm based upon theoretical considerations of hirsch paths seems to be one of the most efficient algorithms for assignment problems since any two basic feasible solutions to a linear problem can always be connected with a short simplex path passing through the infeasible region development of algorithms based upon theoretical considerations on infeasible paths seems to be of great practical interest this paper presents an algorithm of this kind for assignment problems

isotonic median regression a linear programming approach the isotonic median regression problem arises in statistics it is known that the isotonic median regression problem with respect to a complete order may be solved by a pool adjacent violators algorithm in this paper we show that this algorithm is a dual method for solving a linear programming formulation of the problem the linear programming approach provides additional insight into the algorithm as well as a simple proof of its validity we also analyze the computational complexity of the algorithm and discuss its significance from the standpoint of linear programming theory

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

interlacing eigenvalues on some operations of graphs in this paper we focus on some operations of graphs and give a kind of eigenvalue interlacing in terms of the adjacency matrix standard laplacian and normalized laplacian also we explore some applications of this interlacing

a bound on the spectral radius of graphs abstract let g be a simple connected graph with n vertices and m edges let a be the adjacency matrix of g the spectral radius a of a satisfies a 2m n 1 with equality if and only if g is isomorphic to one of the following two graphs a the star k1 n 1 b the complete graph kn

active set algorithms for isotonic regression a unifying framework in this and subsequent papers we will show that several algorithms for the isotonic regression problem may be viewed as active set methods the active set approach provides a unifying framework for studying algorithms for isotonic regression simplifies the exposition of existing algorithms and leads to several new efficient algorithms we also investigate the computational complexity of several algorithms

the spectral radius of bicyclic graphs with prescribed degree sequences restricted to the bicyclic graphs with prescribed degree sequences we determine the unique graph with the largest spectral radius with respect to the adjacency matrix

on the weighted trees with given degree sequence and positive weight set abstract we determine the unique weighted tree with the largest spectral radius with respect to the adjacency and laplacian matrix in the set of all weighted trees with a given degree sequence and positive weight set moreover we also derive the weighted trees with the largest spectral radius with respect to the matrices mentioned above in the sets of all weighted trees with a given maximum degree or pendant vertex number and so on

on the spectral radius of bicyclic graphs with n vertices and diameter d bicyclic graphs are connected graphs in which the number of edges equals the number of vertices plus one in this paper we determine the graph with the largest spectral radius among all bicyclic graphs with n vertices and diameter d as an application we give first three graphs among all bicyclic graphs on n vertices ordered according to their spectral radii in decreasing order

efficient dual simplex algorithms for the assignment problem efficient algorithms based upon balinski s signature method are described for solving then n assignment problem these algorithms are special variants of the dual simplex method and are shown to have computational bounds of o n3 variants for solving sparse assignment problems withm arcs that require o m space and o mn n2 logn time in the worst case are also presented

atlas of finite groups maximal subgroups and ordinary characters for simple groups this atlas covers groups from the families of the classification of finite simple groups recently updated incorporating corrections

the inverse galois problem and rational points on moduli spaces we reduce the regular version of the inverse galois problem for any finite group g to finding one rational point on an infinite sequence of algebraic varieties as a consequence any finite group g is the galois group of an extension l p x with l regular over any pac field p of characteristic zero a special case of this implies that g is a galois group over fp x for almost all primes p

signature classes of transportation polytopes signature algorithms solve certain classes of transportation problems in a number of steps bounded by the diameter of the dual polyhedron the class of problems to which signature algorithms may be applied the signature classes of the title are characterized and the monotonic hirsch conjecture is shown to hold for them in addition certain more precise results are given for different cases finally it is explained why a supposed proof of the hirsch conjecture for all transportation polytopes is incomplete and apparently irremedial

signature methods for the assignment problem the signature of a dual feasible basis of the assignment problem is an n vector whose ith component is the number of nonbasic activities of type i j this paper uses signatures to describe a method for finding optimal assignments that terminates in at most n 1n 2 2 pivot steps and takes at most on3 work

on solving a variation of the assignment problem abstract geetha and nair recently proposed a variant of n n assignment problem with the objective of minimizing the total cost plus an additional supervisory cost they gave an o n 5 algorithm to solve the problem here we present an algorithm for this problem using balinski s signature method for the assignment problem at each stage the algorithm has a complexity of o m m n log n where m is the number of arcs in the problem

new method and new results on the order of spectral radius this paper presents a simple method to order trees unicyclic graphs and bicyclic graphs according to their spectral radii which is based on the application of the majorization theorem of tree unicyclic and bicyclic graphs respectively moreover we obtain some new results on the order of spectral radii of unicyclic and bicyclic graphs by employing this new method

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

laplacian coefficients of trees with given number of leaves or vertices of degree two abstract let g be a simple undirected n vertex graph with the characteristic polynomial of its laplacian matrix l g det i l g k 0 n 1 k c k 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 graph using a result of zhou and gutman on the relation between the laplacian coefficients and the matching numbers in subdivided bipartite graphs we characterize the trees with k leaves pendent vertices which simultaneously minimize all laplacian coefficients in particular this extremal balanced starlike tree s n k minimizes the wiener index the modified hyper wiener index and recently introduced laplacian like energy we prove that graph s n n 1 p has minimal laplacian coefficients among n vertex trees with p vertices of degree two in conclusion we illustrate on examples of these spectrum based invariants that the opposite problem of simultaneously maximizing all laplacian coefficients has no solution and pose a conjecture on extremal unicyclic graphs with k leaves

a competitive dual simplex method for the assignment problem where there is abundance of mystery and confusion in every direction the truth seldom remains hidden for long it s a matter of having plenty of angles to go at it from only the utterly simple crimes the simplex crimes you may say have the trick of remaining baffling sir john from michael innes the open house a sir john appleby mystery penguin books 1974

spectral radius of digraphs with given dichromatic number abstract let d be a digraph with vertex set v d a partition of v d into k acyclic sets is called a k coloring of d the minimum integer k for which there exists a k coloring of d is the dichromatic number d of the digraph d denote g n k the set of the digraphs of order n with the dichromatic number k 2 in this note we characterize the digraph which has the maximal spectral radius in g n k our result generalizes the result of 8 by feng et al

some results on the index of unicyclic graphs we identify in some classes of unicyclic graphs of fixed order and girth those graphs whose index i e the largest eigenvalue is maximal besides some lower and upper bounds on the indices of the graphs being considered are provided

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

the majorization theorem of extremal pseudographs a pseudograph is a graph in which both loops and multiple edges are permitted suppose d1 d2 dn and d1 d2 dn are two positive non increasing degree sequences we write if and only if i 1ndi i 1ndi and i 1jdi i 1jdi for all j 1 2 n let be the class of connected undirected pseudographs with degree sequence let g and g be the spectral radius and signless laplacian spectral radius of g respectively in this paper the extremal pseudographs with the largest respectively signless laplacian spectral radii in are characterized furthermore we show that if g and g are the pseudographs with the largest respectively signless laplacian spectral radii in and respectively then g g and g g