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

aα spectral radius of the second power of a graph abstract the kth power of a graph g denoted by gk is a graph with the same set of vertices as g such that two vertices are adjacent in gk if and only if their distance in g is at most k in this paper we give upper and lower bounds on the a spectral radius of g2 furthermore the first three largest a spectral radius of the second power of trees are determined

the cutting of compositions this paper investigates the cutability of the fuzzy relational compositions the sup t composition the triangular compositions of bandler and kohout and their improved versions the theoretical and practical importance of fuzzy cuts is illustrated by means of a careful selection of results from the fuzzy literature some new potential properties of fuzzy implications operators and triangular norms that play an important role in this cutability study are introduced and discussed extensively the paper concludes with a study of the fuzzy cutability of the sup t composition

an explicit formula for eigenvalues of bethe trees and upper bounds on the largest eigenvalue of any tree a bethe tree bd k is a rooted unweighted of k levels in which the root vertex has degree equal to d the vertices at level j 2 j k 1 have degree equal to d 1 and the vertices at level k are the pendant vertices in this paper we first derive an explicit formula for the eigenvalues of the adjacency matrix of bd k moreover we give the corresponding multiplicities next we derive an explicit formula for the simple nonzero eigenvalues among them the largest eigenvalue of the laplacian matrix of bd k finally we obtain upper bounds on the largest eigenvalue of the adjacency matrix and of the laplacian matrix of any tree t these upper bounds are given in terms of the largest vertex degree and the radius of t and they are attained if and only if t is a bethe tree

relational duals of the c means clustering algorithms abstract the hard and fuzzy c means algorithms are widely used effective tools for the problem of clustering n objects into hard or fuzzy groups of similar individuals when the data is available as object data consisting of a set of n feature vectors in rp however object data algorithms are not directly applicable when the n objects are implicitly described in terms of relational data which consists of a set of n2 measurements of relations between each of the pairs of objects new relational versions of the hard and fuzzy c means algorithms are presented here for the case when the relational data can reasonably be viewed as some measure of distance some convergence properties of the algorithms are given along with a numerical example

the aα spectral radius characterization of some digraphs abstract let d be the a spectral radius of digraph d and let g n r be the set of digraphs with order n and dichromatic number r in this paper we characterize the digraph which has the maximal a spectral radius in g n r moreover we also determine the unique digraph which attains the maximum resp minimum a spectral radius among all strongly connected bicyclic digraphs

on the aα spectral radius of a graph 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 3 defined the matrix a g as a g d g 1 a g the largest eigenvalue of a g is called the a spectral radius of g in this paper we give three edge graft transformations on a spectral radius as applications we determine the unique graph with maximum a spectral radius among all connected graphs with diameter d and determine the unique graph with minimum a spectral radius among all connected graphs with given clique number in addition some bounds on the a spectral radius are obtained

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

an analysis of some fuzzy subset applications to information retrieval systems abstract several authors have attempted to improve the ability of an information retrieval system to process queries by allowing for a more general query and query processing structure than standard boolean expressions and ordinary set theory the first step in generalization from ordinary set theory to fuzzy subset theory is both philosophically and mathematically straightforward the next step that of generalizing the use of fuzzy subset theory has been more difficult of particular fundamental concern is the underlying lattice structure of the fuzzy subsets themselves and the possible loss of certain features of that structure in the second generalization step this paper will show that the mathematical requirements placed on the generalization in order that the lattice structures be preserved are really quite minimal

on the multiplicity of an arbitrary aα eigenvalue of a connected graph abstract the a matrix of a graph g is defined by nikiforov as a g d g 1 a g where 0 1 a g and d g respectively denotes the adjacency matrix and the degree diagonal matrix of g the eigenvalues of a g are called the a eigenvalues of g in this paper we prove that for a connected graph g of order n and with maximum degree 2 and for any 0 1 the multiplicity of an arbitrary a eigenvalue of g which is denoted as m g is bounded by m g 2 n 2 1 the equality holds if and only if g and satisfy one of the following conditions i g k n with n 3 and n 1 ii g c n with even order and 2 2 1 cos 2 j n j 1 2 n 2 2 iii g c n with odd order and 2 2 1 cos 2 j n j 1 2 n 1 2 iv g k n 2 n 2 with n 4 and n 2 applying this result we solve an open conjecture posed by zhou et al in 32 one year ago

on the multiplicity of α as an eigenvalue of the aα matrix of a graph in terms of the number of pendant vertices abstract let g v g e g be a simple undirected graph with vertex set v g and edge set e g the cyclomatic number of a connected graph g is defined as g e g v g 1 the a matrix of a graph g is defined by nikiforov as a g d g 1 a g where 0 1 a g and d g respectively denotes the adjacency matrix and the diagonal matrix of the vertex degrees of g let m g be the multiplicity of as an eigenvalue of a g a cluster of g is an independent set of one or more vertices of g each of which has the same set of neighbors the closure of a cluster is the set of vertices of the cluster together with all the neighbors of the cluster the degree of a cluster is the cardinality of its shared set of neighbors a k cluster is a cluster of degree k the number of vertices in a k cluster is its order a collection of two or more k clusters is independent if the k clusters are pairwise disjoint and is isolated if the closures of the k clusters are pairwise disjoint cardoso et al 2 obtained a lower bound for m g in terms of the number of pendant vertices they proved that for a simple connected graph g with p g 0 pendant vertices attached at q g quasi pendant vertices m g p g q g with equality if each internal vertex is a quasi pendant vertex in this article for a graph g of order n and with t independent k clusters of orders r 1 r t we prove that m g k i 1 t r i t and if the k clusters are isolated then m g k n 2 t which extends cardoso s result to a more general case also we give an upper bound for m g in terms of p g it is proved that m g p g 2 g with g the cyclomatic number of g the graphs g whose m g attain the upper bound are characterized explicitly

on the aα characteristic polynomial of a graph abstract let g be a graph with n vertices and let a g and d g denote respectively the adjacency matrix and the degree matrix of g define a g d g 1 a g for any real 0 1 the a characteristic polynomial of g is defined to be det x i n a g j c j g x n j where det denotes the determinant of and i n is the identity matrix of size n the a spectrum of g consists of all roots of the a characteristic polynomial of g a graph g is said to be determined by its a spectrum if all graphs having the same a spectrum as g are isomorphic to g in this paper we first formulate the first four coefficients c 0 g c 1 g c 2 g and c 3 g of the a characteristic polynomial of g and then we observe that a spectra are much efficient for us to distinguish graphs by enumerating the a characteristic polynomials for all graphs on at most 10 vertices to verify this observation we characterize some graphs determined by their a spectra

the spectra of the adjacency matrix and laplacian matrix for some balanced trees abstract let t be an unweighted rooted tree of k levels such that in each level the vertices have equal degree let dk j 1 denotes the degree of the vertices in the level j we find the eigenvalues of the adjacency matrix and of the laplacian matrix of t they are the eigenvalues of principal submatrices of two nonnegative symmetric tridiagonal matrices of order k k the codiagonal entries for both matrices are d j 1 2 j k 1 and d k while the diagonal entries are zeros in the case of the adjacency matrix and dj 1 j k in the case of the laplacian matrix moreover we give some results concerning to the multiplicity of the above mentioned eigenvalues

a new approach to information retrieval systems using fuzzy expressions a fuzzy document retrieval system has been realized that permits a reduction of the time needed for retrieving the set of documents fulfilling a user s query as well as the amount of core space needed for the document description relations the method described solves the problems that radecki who uses lambda level fuzzy sets met with trying to reduce time and gives rise to a good working information retrieval system as the examples show the retrieval process is described by an included algorithm

applications of estrada indices and energy to a family of compound graphs abstract to track the gradual change of the adjacency matrix of a simple graph g into the signless laplacian matrix v nikiforov in 35 suggested the study of the convex linear combination a adjacency matrix a g d g 1 a g for 0 1 where a g and d g are the adjacency and the diagonal vertex degrees matrices of g respectively taking this definition as an idea the next matrix was considered for a b r the matrix a a b defined by a a b g a d g b a g extends the previous adjacency matrix this matrix is designated the a b adjacency matrix of g both adjacency matrices are examples of universal matrices already studied by w haemers in this paper we study the a b adjacency spectra for a family of compound graphs formed by disjoint balanced trees whose roots are identified to the vertices of a given graph in consequence new families of cospectral adjacency laplacian and signless laplacian graphs new hypoenergetic graphs graphs whose energy is less than its vertex number and new explicit formulae for estrada signless laplacian estrada and laplacian estrada indices of graphs were obtained moreover sharp upper bounds of the above indices for caterpillars in terms of length of the path and of the maximum number of its pendant vertices are given

the aα spread of a graph abstract let a g and d g be the adjacency matrix and the degree diagonal matrix of a graph g respectively for any real number 0 1 nikiforov defined the a matrix of a graph g as a g d g 1 a g the a spread of a graph is the difference between the largest eigenvalue and the smallest eigenvalue of the a matrix of the graph in this paper we obtain some new lower bounds on a spread and characterize the extremal graphs for certain cases which extend the results of a spread and q spread moreover some graph operations on a spread are presented

optimal transport from lebesgue to poisson this paper is devoted to the study of couplings of the lebesgue measure and the poisson point process we prove existence and uniqueness of an optimal coupling whenever the asymptotic mean transportation cost is finite moreover we give precise conditions for the latter which demonstrate a sharp threshold at d 2 the cost will be defined in terms of an arbitrary increasing function of the distance the coupling will be realized by means of a transport map allocation map which assigns to each poisson point a set cell of lebesgue measure 1 in the case of quadratic costs all these cells will be convex polytopes

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

bounds for the largest and the smallest aα eigenvalues of a graph in terms of vertex degrees abstract let g be a graph with adjacency matrix a g and with d g the diagonal matrix of its vertex degrees nikiforov defined the matrix a g with 0 1 as a g d g 1 a g the largest and the smallest eigenvalues of a g are respectively denoted by g and n g in this paper we present a tight upper bound for g in terms of the vertex degrees of g for 1 for any graph g g max u w d u d w 2 d u d w 2 4 1 2 d u d w 2 if g is connected for 0 1 the equality holds if and only if g is regular or bipartite semi regular as an application we solve the problem raised by nikiforov in 15 for the smallest eigenvalue n g of a bipartite graph g of order n with no isolated vertices for 0 1 then n g min u w d u d w 2 d u d w 2 4 1 2 d u d w 2 if g is connected and 1 2 the equality holds if and only if g is regular or semi regular

on the α index of graphs with pendent paths abstract let g be a graph with adjacency matrix a g and let d g be the diagonal matrix of the degrees of g for every real 0 1 write a g for the matrix a g d g 1 a g this paper presents some extremal results about the spectral radius g of a g that generalize previous results about 0 g and 1 2 g in particular write b p q r be the graph obtained from a complete graph k p by deleting an edge and attaching paths p q and p r to its ends it is shown that if 0 1 and g is a graph of order n and diameter at least k then g b n k 2 k 2 k 2 with equality holding if and only if g b n k 2 k 2 k 2 this result generalizes results of hansen and stevanovic 5 and liu and lu 7

a fuzzy document retrieval method based on two valued indexing abstract a fuzzy retrieval method that enables one to infer weights and ranking outputs is formulated in this paper still based on two valued indexing it provides a weighted system of document retrieval a keyword is represented by a set of keywords related to it as a document then a weight is calculated by the degree of matching between the representation of a keyword and that of a document this procedure can obviously be extended to boolean queries by fuzzy logic and hence to weighted boolean systems without assuming weighted indexing an interpretation of this method from the point of view of the possible world semantics in modal logic is further discussed

on the multiplicity of α as an eigenvalue of aα g of graphs with pendant vertices abstract let g be a simple undirected graph let 0 1 let a g d g 1 a g where d g and a g are the diagonal matrix of the vertex degrees of g and the adjacency matrix of g respectively let p g 0 and q g be the number of pendant vertices and quasi pendant vertices of g respectively let m g be the multiplicity of as an eigenvalue of a g it is proved that m g p g q g with equality if each internal vertex is a quasi pendant vertex if there is at least one internal vertex which is not a quasi pendant vertex the equality m g p g q g m n is determined in which m n is the multiplicity of as an eigenvalue of the matrix n this matrix is obtained from a g taking the entries corresponding to the internal vertices which are non quasi pendant vertices these results are applied to search for the multiplicity of as an eigenvalue of a g when g is a path a caterpillar a circular caterpillar a generalized bethe tree or a bethe tree for the bethe tree case a simple formula for the nullity is given