the spectral radius of tricyclic graphs with n vertices and k pendent vertices in this paper we determine graphs with the largest spectral radius among all the tricyclic graphs with n vertices and k pendent vertices

ordering of some uniform supertrees with larger spectral radii abstract a supertree is a connected and acyclic hypergraph we study some uniform supertrees with larger spectral radii we first define a new type of edge moving operation on hypergraphs and study its applications on the comparison of the spectral radii of hypergraphs by using this new operation on some supertrees together with the general edge moving operation introduced by li shao and qi in 2015 and using a result by zhou et al in 2014 about the relation between the spectral radii of an ordinary graph and its power hypergraph we are able to determine the first eight k uniform supertrees on n vertices with the largest spectral radii

on the spectral radius of weighted trees with fixed diameter and weight set abstract the spectrum of weighted graphs are often used to solve the problems in the design of networks and electronic circuits we first give some perturbational results on the spectral radius of weighted graphs when some weights of edges are modified then we derive the weighted tree with the largest spectral radius in the set of all weighted trees with fixed diameter and weight set furthermore an open problem of spectral radius on weighted paths is solved h z yang g z hu y hong bounds of spectral radii of weighted tree tsinghua science and technology 8 2003 517 520

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

extension d une valuation we want to determine all the extensions of a valuation v of a field k to a cyclic extension l of k i e l k x is the field of rational functions of x or l k is the finite separable extension generated by a root 0 of an irreducible polynomial g x in two articles from 1936 saunders maclane has introduced the notions of key polynomial and of augmented valuation for a given valuation of k x and has shown how we can recover any extension to l of a discrete rank one valuation v of k by a countable sequence of augmented valuations with i n the valuation i is defined by induction from the valuation i 1 from a key polynomial o i and from the value oi in this article we study some properties of the augmented valuations and we generalize the results of maclane to the case of any valuation v of k for this we need to introduce simple admissible families of augmented valuations a a where a is not necessarily a countable set and to define a limit key polynomial and limit augmented valuation for such families then any extension to l of a valuation on k is again a limit of a family of augmented valuations we also get a factorization theorem which gives a description of the values f for any polynomial f in k x

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

indexing hierarchical structures using graph spectra hierarchical image structures are abundant in computer vision and have been used to encode part structure scale spaces and a variety of multiresolution features in this paper we describe a framework for indexing such representations that embeds the topological structure of a directed acyclic graph dag into a low dimensional vector space based on a novel spectral characterization of a dag this topological signature allows us to efficiently retrieve a promising set of candidates from a database of models using a simple nearest neighbor search we establish the insensitivity of the signature to minor perturbation of graph structure due to noise occlusion or node split merge to accommodate large scale occlusion the dag rooted at each nonleaf node of the query votes for model objects that share that part effectively accumulating local evidence in a model dag s topological subspaces we demonstrate the approach with a series of indexing experiments in the domain of view based 3d object recognition using shock graphs

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

key polynomials and pseudo convergent sequences abstract in this paper we introduce a new concept of key polynomials for a given valuation on k x we prove that such polynomials have many of the expected properties of key polynomials as those defined by maclane and vaquie for instance that they are irreducible and that the truncation of associated to each key polynomial is a valuation moreover we prove that every valuation on k x admits a sequence of key polynomials that completely determines in the sense which we make precise in the paper we also establish the relation between these key polynomials and pseudo convergent sequences defined by kaplansky

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

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

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

sharp upper bounds on the second largest eigenvalues of connected graphs abstract let 2 be the second largest eigenvalue of a graph powers 1988 4 gave some upper bounds of 2 for general graphs and bipartite graphs respectively considering that these bounds are not always attainable for connected graphs we present sharp upper bounds of 2 for connected graphs and connected bipartite graphs in this paper moreover the extremal graphs are completely characterized

every place admits local uniformization in a finite extension of the function field we prove that every place p of an algebraic function field f k of arbitrary characteristic admits local uniformization in a finite extension f of f we show that f f can be chosen to be galois after a finite purely inseparable extension of the ground field k instead of being galois the extension can also be chosen such that the induced extension fp fp of the residue fields is purely inseparable and the value group of f only gets divided by the residue characteristic if f lies in the completion of an abhyankar place then no extension of f is needed our proofs are based solely on valuation theoretical theorems which are of particular importance in positive characteristic they are also applicable when working over a subring r k and yield similar results if r is regular and of dimension smaller than 3

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

the minimum spectral radius of graphs with a given independence number abstract the independence number g of g is defined as the maximum cardinality of a set of pairwise non adjacent vertices which is called an independent set in this paper we characterize the graphs which have the minimum spectral radius among all the connected graphs of order n with independence number 1 2 n 2 n 2 1 n 3 n 2 n 1

w tsv weighted topological signature vector for lexicon reduction in handwritten arabic documents this paper proposes a holistic lexicon reduction method for ancient and modern handwritten arabic documents the word shape is represented by the weighted topological signature vector w tsv which encodes graph data into a low dimensional vector space three directed acyclic graph dag representations are proposed for arabic word shapes based on topological and geometrical features lexicon reduction is achieved by a nearest neighbors search in the w tsv space the proposed framework has been tested on the ifn enit and the ibn sina databases achieving respectively a degree of reduction of 83 5 and 92 9 for an accuracy of reduction of 90

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

skew adjacency matrices of graphs the spectra of the skew adjacency matrices of a graph are considered as a possible way to distinguish adjacency cospectral graphs this leads to the following topics graphs whose skew adjacency matrices are all cospectral relations between the matchings polynomial of a graph and the characteristic polynomials of its adjacency and skew adjacency matrices skew spectral radii and an analogue of the perron frobenius theorem and the number of skew adjacency matrices of a graph with distinct spectra

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

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

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

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 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

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

indices for special classes of trees abstract in this article we study the index the largest eigenvalue of the adjacency matrix in two special classes of trees namely starlike trees and double brooms for each class we determine conditions for the index to be integer we examine conditions in order to compare the index of a starlike tree with 1 where is the maximum degree of the graph we also prove that indices of a broom like tree and a double broom except for some cases are between and 1 characterizing when occurs one equality furthermore we build in these classes infinite families of non integral trees with integer index

notes on the second largest eigenvalue of a graph abstract for a fixed real number r we give several necessary and or sufficient conditions for a graph to have the second largest eigenvalue of the adjacency matrix or signless laplacian matrix less then or equal to r

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

a conjecture on the second largest eigenvalue of a tree with perfect matchings abstract in this paper we present an upper bound for the second largest eigenvalue of a tree on n 2 k 4 t t 2 vertices with perfect matchings at the same time the few largest second largest eigenvalues over the class of trees on n 2 k 4 t t 2 vertices with perfect matchings are also obtained

a note on the spectral characterization of strongly connected bicyclic digraphs let d be a digraph with vertex set v d and a be the adjacency matrix of d in this paper we characterize the extremal digraphs which achieve the maximum and minimum spectral radius among strongly connected bicyclic digraphs furthermore we show that any strongly connected bicyclic digraph is determined by the spectrum

key polynomials and minimal pairs abstract in this paper we establish the relation between key polynomials as defined in 12 and minimal pairs of definition of a valuation we also discuss truncations of valuations on a polynomial ring k x we prove that a valuation is equal to its truncation on some polynomial if and only if is valuation transcendental another important result of this paper is that if is any extension of to k x and is a complete sequence of key polynomials for without last element then for each q there exists a suitable root a q k of q such that a q q is a pseudo convergent sequence defining

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

sharp upper bounds on the spectral radius of graphs abstract let g be a simple connected graph with n vertices m edges and degree sequence d 1 d 2 d n the spectral radius g of graph g is the largest eigenvalue of its adjacency matrix in this paper we present some sharp upper bounds of the spectral radius in terms of the degree sequence of graphs

bounds on graph eigenvalues abstract upper and lower estimates are found for the maximum of the kth eigenvalue of a graph as a function of the number of vertices or edges

resolution of singularities of threefolds in positive characteristic ii abstract together with vincent cossart olivier piltant resolution of singularities of threefolds in positive characteristic i reduction to local uniformization on artin schreier and purely inseparable coverings j algebra 320 3 2008 1051 1082 this article gives a complete proof of desingularization of quasiprojective varieties of dimensional 3 on fields which are differentially finite over perfect fields

on the spectra of some weighted rooted trees and applications abstract let t be a weighted rooted tree of k levels such that 1 the vertices in level j have a degree equal to d k j 1 for j 1 2 k and 2 the edges joining the vertices in level j with the vertices in level j 1 have a weight equal to w k j for j 1 2 k 1 we give a complete characterization of the eigenvalues of the laplacian matrix and adjacency matrix of t they are the eigenvalues of leading principal submatrices of two nonnegative symmetric tridiagonal matrices of order k k moreover we give some results concerning their multiplicities by application of the above mentioned results we derive upper bounds on the largest eigenvalue of any weighted tree and the spectra of some weighted bethe trees

on the spectral radius of trees with fixed diameter let t n d be the set of trees on n vertices with diameter d in this paper the first d2 1 spectral radii of trees in the set t n d 3 d n 4 are characterized

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

the kth largest eigenvalue of a tree abstract let k t be the k th eigenvalue of a tree x the largest integer not greater than x it is shown that a tree with n vertices has k t n 2 k for 2 k n 2 and this upper bound is best possible for n 1 mod k

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

maximal reflexive cacti with four cycles the approach via smith graphs abstract cacti or treelike graphs are graphs whose all cycles are mutually edge disjoint graphs with the property 2 2 are called reflexive graphs where 2 is the second largest eigenvalue of the corresponding 0 1 adjacency matrix the property 2 2 is a hereditary one i e all induced subgraphs of a reflexive graph are also reflexive this is why we represent reflexive graphs through the maximal graphs within a given class such as connected cacti with a fixed number of cycles in previous work we have determined all maximal reflexive cacti with four cycles whose cycles do not form a bundle and pointed out the role of so called pouring of smith graphs in their construction in this paper besides pouring we show several other patterns of the appearance of smith trees in those constructions these include splitting of a smith tree adding an edge to a smith tree and then splitting of the resulting graph identifying two vertices of a smith tree and then splitting the resulting graph our results show that the presence of smith trees is evident in all such maximal reflexive cacti with four cycles and that in most of them smith graphs appear in the described way

modular lie algebras and the gelfand kirillov conjecture let mathfrak g be a finite dimensional simple lie algebra over an algebraically closed field mathbb k of characteristic 0 let mathfrak g mathbb z be a chevalley form of mathfrak g and mathfrak g bbbk mathfrak g mathbb z otimes mathbb z bbbk where bbbk is the algebraic closure of mathbb f p let g bbbk be a simple simply connected algebraic bbbk group with operatorname lie g bbbk mathfrak g bbbk in this paper we apply recent results of rudolf tange on the fraction field of the centre of the universal enveloping algebra u mathfrak g bbbk to show that if the gelfand kirillov conjecture from 1966 holds for mathfrak g then for all p 0 the field of rational functions bbbk mathfrak g bbbk is purely transcendental over its subfield bbbk mathfrak g bbbk g bbbk very recently it was proved by colliot thelene kunyavski popov and reichstein that the field of rational functions mathbb k mathfrak g is not purely transcendental over its subfield mathbb k mathfrak g mathfrak g if mathfrak g is of type b n n 3 d n n 4 e6 e7 e8 or f4 we prove a modular version of this result valid for p 0 and use it to show that in characteristic 0 the gelfand kirillov conjecture fails for the simple lie algebras of the above types in other words if mathfrak g is of type b n n 3 d n n 4 e6 e7 e8 or f4 then the lie field of mathfrak g is more complicated than expected

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

a 1 0 1 and sparsest basis for the null space of a forest in optimal time abstract given a matrix the null space problem asks for a basis of its null space having the fewest nonzeros this problem is known to be np complete and even hard to approximate the null space of a forest is the null space of its adjacency matrix sander and sander 2005 and akbari et al 2006 independently proved that the null space of each forest admits a 1 0 1 basis we devise an algorithm for determining a sparsest basis of the null space of any given forest which in addition is a 1 0 1 basis our algorithm is time optimal in the sense that it takes time at most proportional to the number of nonzeros in any sparsest basis of the null space of the input forest moreover we show that given a forest f on n vertices the set of those vertices x for which there is a vector in the null space of f that is nonzero at x and the number of nonzeros in any sparsest basis of the null space of f can be found in o n time

gelfand kirillov conjecture in positive characteristics abstract let g be a finite dimensional lie algebra over an algebraically closed field of characteristic p 0 and let u g be the universal enveloping algebra of g we show in this paper that the division ring of fractions of u g is isomorphic to the ring of fractions of a weyl algebra in the following cases for g gl n or sl n if p n for the witt algebra w 1 and for some tensor product w 1 a of w 1 with a truncated polynomial ring furthermore we also show that the centre of u g in the last two cases is a unique factorisation domain in accordance with recent results of premet tange braun and hajarnavis

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

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

on the spectra of certain rooted trees abstract let t be an unweighted tree with vertex root v which is the union of two trees t 1 v 1 e 1 t 2 v 2 e 2 such that v 1 v 2 v and t 1 and t 2 have the property that the vertices in each of their levels have equal degree we characterize completely the eigenvalues of the adjacency matrix and of the laplacian matrix of t they are the eigenvalues of symmetric tridiagonal matrices whose entries are given in terms of the vertex degrees moreover we give some results about the multiplicity of the eigenvalues applications to some particular trees are developed

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

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

on the largest kth eigenvalues of trees abstract we consider the only remaining unsolved case n 0 mod k for the largest k th eigenvalue of trees with n vertices we give complete solutions for the cases k 2 3 4 5 and give some necessary conditions for extremal trees in general cases

the spectral radius of unicyclic and bicyclic graphs with n vertices and k pendant vertices in this paper we determine graphs with the largest spectral radius among all the unicyclic and all the bicyclic graphs with n vertices and k pendant vertices respectively

field of fractions of certain quantum algebras abstract in this paper we show that the field of fractions of certain algebras appearing in the theory of quantum groups are isomorphic to the field of fractions of quantum n space when the parameter q is not a root of 1 more precisely the quantum serre relations of type a n and the relations defining the quantum weyl algebra induce q communtation relations in the field of fractions also we study the problem of separation of these skew fields using the standard embeddings in non commutative series

the graphs with spectral radius between 2 and 2 5 abstract we complete the determination of the graphs in the title begun by cvetkovic doob and gutman

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

a study of eigenspaces of graphs abstract we investigate the relationship between the structure of a graph and its eigenspaces the angles between the eigenspaces and the vectors of the standard basis of r n play an important role the key notion is that of a special basis for an eigenspace called a star basis star bases enable us to define a canonical bases of r n associated with a graph and to formulate an algorithm for graph isomorphism

effacement des derivations et spectres premiers des algebres quantiques abstract given any commutative field k and any iterated ore extension r k x1 x2 2 2 xn n n satisfying some suitable assumptions we construct the so called derivative elimination algorithm it consists of a sequence of changes of variables inside the division ring f fract r starting with the indeterminates x1 xn and terminating with new variables t1 tn these new variables generate some quantum affine space r such that f fract r this algorithm induces a natural embedding spec r spec r which satisfies the following property if p is in spec r then fract r p fract r p we study both the derivative elimination algorithm and natural embedding and use them to produce for the general case a common proof of the quantum gelfand kirillov property for the prime homomorphic images of the following quantum algebras u q w g b q w g w w r q g where g denotes any complex semi simple connected simply connected lie group with associated lie algebra g and weyl group w quantum matrices algebras quantum weyl algebras and quantum euclidean respectively symplectic spaces another application will be given in g cauchon j algebra to appear in the general case the prime spectrum of any quantum matrices algebra satisfies the normal separation property

on the spectral radius of graphs with cut edges we study the spectral radius of graphs with n vertices and k cut edges in this paper we show that of all the connected graphs with n vertices and k cut edges the maximal spectral radius is obtained uniquely at k k where k k n is a graph obtained by joining k independent vertices to one vertex of kn k we also discuss the limit point of the maximal spectral radius

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

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 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

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

pi degree parity in q skew polynomial rings abstract for k a field of arbitrary characteristic and r a k algebra we show that the pi degree of an iterated skew polynomial ring r x 1 1 1 x n n n agrees with the pi degree of r x 1 1 x n n when each i i satisfies a q i skew relation for q i k and extends to a higher q i skew i derivation we confirm the quantum gel fand kirillov conjecture for various quantized coordinate rings and calculate their pi degrees we extend these results to completely prime factor algebras

on the spectral radius of tricyclic graphs with a maximum matching abstract in this paper we determine the graph with the largest spectral radius among all the tricyclic graphs with a maximum matching of cardinality m m 8

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 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

ordering trees by their largest eigenvalues abstract the set of trees with n vertices is denoted by t n hofmeister has determined the first five values of the largest eigenvalue of trees in t n and the corresponding trees for these values linear algebra appl 260 1997 43 in other words an order of the first five trees in t n by their largest eigenvalues has been given focus on the same purpose we shall give a partition for trees in t n first in the this paper and then extend this order to the eighth tree

ordering trees by their largest eigenvalues abstract let t and 1 t denote the maximum degree and the largest eigenvalue of a tree t respectively let t n be the set of trees on n vertices and t n t t n t in the present paper among the trees in t n n 4 we characterize the tree which alone minimizes the largest eigenvalue as well as the tree which alone maximizes the largest eigenvalue when n 2 2 n 1 furthermore it is proved that for two trees t1 and t2 in t n n 4 if t 1 2 n 3 1 and t1 t2 then 1 t1 1 t2 by applying this result we extend the order of trees in t n by their largest eigenvalues to the 13th tree when n 12 this extends the results of hofmeister linear algebra appl 260 1997 43 and chang et al linear algebra appl 370 2003 175

a class of counterexamples to the gel fand kirillov conjecture limburgs univ centrum dept wni b 3500 diepenbeek belgium alev j univ reims ufr sci dept math moulin housse bp 347 f 51062 reims france

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

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

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

value groups residue fields and bad places of rational function fields we classify all possible extensions of a valuation from a ground field k to a rational function field in one or several variables over k we determine which value groups and residue fields can appear and we show how to construct extensions having these value groups and residue fields in particular we give several constructions of extensions whose corresponding value group and residue field extensions are not finitely generated in the case of a rational function field k x in one variable we consider the relative algebraic closure of k in the henselization of k x with respect to the given extension and we show that this can be any countably generated separable algebraic extension of k in the tame case we show how to determine this relative algebraic closure finally we apply our methods to power series fields and the p adics

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 graphs whose signless laplacian index does not exceed 4 5 abstract let a g and d g be respectively the adjacency matrix and the degree matrix of a graph g the signless laplacian matrix of g is defined as q g d g a g the q spectrum of g is the set of the eigenvalues together with their multiplicities of q g the q index of g is the maximum eigenvalue of q g the possibilities for developing a spectral theory of graphs based on the signless laplacian matrices were discussed by cvetkovic et al d cvetkovic p rowlinson s k simic signless laplacians of finite graphs linear algebra appl 423 2007 155 171 in the latter paper the authors determine the graphs whose q index is in the interval 0 4 in this paper we investigate some properties of q spectra of graphs especially for the limit points of the q index by using these results we characterize respectively the structures of graphs whose the q index lies in the intervals 4 2 5 2 5 2 and 2 4 5 where 1 3 54 6 33 1 3 54 6 33 1 3 2 382975767

resolution of singularities of threefolds in positive characteristic i reduction to local uniformization on artin schreier and purely inseparable coverings abstract in dimension 3 the desingularization problem is reduced to local uniformization on artin schreier and purely inseparable coverings of regular space

a quadratic poisson gel fand kirillov problem in prime characteristic the quadratic poisson gel fand kirillov problem asks whether the field of fractions of a poisson algebra is poisson birationally equivalent to a poisson affine space i e to a polynomial algebra k x1 xn with poisson bracket defined by xi xj ijxixj for some skew symmetric matrix ij mn k this problem was studied in 9 over a field of characteristic 0 by using a poisson version of the deleting derivation homomorphism of cauchon in this paper we study the quadratic poisson gel fand kirillov problem over a field of arbitrary characteristic in particular we prove that the quadratic poisson gel fand kirillov problem is satisfied for a large class of poisson algebras arising as semiclassical limits of quantised coordinate rings for we introduce the concept of higher poisson derivation which allows us to extend the poisson version of the deleting derivation homomorphism from the characteristic 0 case to the case of arbitrary characteristic when a torus is acting rationally by poisson automorphisms on a poisson polynomial algebra arising as the semiclassical limit of a quantised coordinate ring we prove under some technical assumptions that quotients by poisson prime torus invariant ideals also satisfy the quadratic poisson gel fand kirillov problem in particular we show that coordinate rings of determinantal varieties satisfy the quadratic poisson gel fand kirillov problem