some further results on the eccentric distance sum abstract let g v g e g be a simple connected graph then the eccentric distance sum of g which is a novel graph invariant by offering a great potential for structure activity property relationships is defined as d g v v e g v d g v where e g v is the eccentricity of the vertex v and d g v is the sum of all distance from the vertex v as a continuation to the parts of 22 li et al j math anal appl 43 1149 1162 2015 and 17 hua et al discrete appl math 160 170 180 2012 this paper answers one of some remaining problems in 22 is how to determine the bipartite graphs of even diameter with the minimum eds and gives a nordhaus gaddum bound for eccentric distance sum which is the generalization of the corresponding results in 17 moreover some sharp lower bounds on the eds of halin graphs and of triangle free and quadrangle free graphs are respectively presented

on the minimum eccentric distance sum of bipartite graphs with some given parameters the eccentric distance sum is a novel graph invariant with vast potential in structure activity property relationships this graph invariant displays high discriminating power with respect to both biological activity and physical properties if g vg eg is a simple connected graph then the eccentric distance sum eds of g is defined as d g v vgeg v dg v where eg v is the eccentricity of the vertex v and dg v u vgdg u v is the sum of all distances from the vertex v much extremal work has been done on trees with some given parameters by yu et al 2011 25 li et al 2012 19 and geng et al 2013 6 it is natural to consider this extremal problem on bipartite graphs with some given parameters in this paper sharp lower bound on the eds in the class of all connected bipartite graphs with a given matching number q is determined the minimum eds is realized only by the graph kq n q the extremal graph with the minimum eds in the class of all the n vertex connected bipartite graphs of odd diameter is characterized all the extremal graphs having the minimum eds in the class of all connected n vertex bipartite graphs with a given vertex connectivity are identified as well

distribution of some graph invariants over hierarchical product of graphs the hierarchical product of graphs was introduced very recently by l barriere et al in on the spectra of hypertrees linear algebra appl 428 2008 1499 1510 and some of its main properties were studied in this paper some new properties of this new graph product are investigated we prove that g n g 1 is median graph if and only if g 1 g 2 g n are median an exact formula for metric dimension of g n g 1 as well as formulas for the eccentric distance sum and edge revised szeged of hierarchical product of graphs are presented some applications of our results are also presented

simplicity transformations for three way arrays with symmetric slices and applications to tucker 3 models with sparse core arrays abstract tucker three way pca and candecomp parafac are two well known methods of generalizing principal component analysis to three way data candecomp parafac yields component matrices a e g for subjects or objects b e g for variables and c e g for occasions that are typically unique up to jointly permuting and rescaling columns tucker 3 analysis on the other hand has full transformational freedom that is the fit does not change when a b and c are postmultiplied by nonsingular transformation matrices provided that the inverse transformations are applied to the so called core array g this freedom of transformation can be used to create a simple structure in a b c and or in g this paper deals with the latter possibility exclusively it revolves around the question of how a core array or in fact any three way array can be transformed to have a maximum number of zero elements direct applications are in tucker 3 analysis where simplicity of the core may facilitate the interpretation of a tucker 3 solution and in constrained tucker 3 analysis where hypotheses involving sparse cores are taken into account in the latter cases it is important to know what degree of sparseness can be attained as a tautology by using the transformational freedom in addition simplicity transformations have proven useful as a mathematical tool to examine rank and generic or typical rank of three way arrays so far a number of simplicity results have been attained pertaining to arrays sampled randomly from continuous distributions these results do not apply to three way arrays with symmetric slices in one direction the present paper offers a number of simplicity results for arrays with symmetric slices of order 2 2 3 3 and 4 4 some generalizations to higher orders are also discussed as a mathematical application the problem of determining the typical rank of 4 3 3 and 5 3 3 arrays with symmetric slices will be revisited using a sparse form with only 8 out of 36 elements nonzero for the former case and 10 out of 45 elements nonzero for the latter one that can be attained almost surely for such arrays the issue of maximal simplicity of the targets to be presented will be addressed either by formal proofs or by relying on simulation results

on the eccentric distance sum of trees and unicyclic graphs abstract let g be a simple connected graph with the vertex set v g the eccentric distance sum of g is defined as d g v v g e v d g v where e v is the eccentricity of the vertex v and d g v u v g d u v is the sum of all distances from the vertex v in this paper we characterize the extremal unicyclic graphs among n vertex unicyclic graphs with given girth having the minimal and second minimal eccentric distance sum in addition we characterize the extremal trees with given diameter and minimal eccentric distance sum

degeneracy in candecomp parafac explained for p p 2 arrays of rank p 1 or higher the candecomp parafac cp model decomposes a three way array into a prespecified number r of rank 1 arrays and a residual array in which the sum of squares of the residual array is minimized the practical use of cp is sometimes complicated by the occurrence of so called degenerate solutions in which some components are highly correlated in all three modes and the elements of these components become arbitrarily large we consider the real valued cp model in which p p 2 arrays of rank p 1 or higher are decomposed into p rank 1 arrays and a residual array it is shown that the cp objective function does not have a minimum in these cases but an infimum moreover any sequence of cp approximations of which the objective value approaches the infimum will become degenerate this result extends ten berge kiers de leeuw 1988 who consider a particular 2 2 2 array of rank 3

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

a bilinear extension inequality in two dimensions abstract we provide sharp decay estimates for circular averages of a certain bilinear extension operator on l 2 s 1 l 2 s 1

on the eccentric distance sum of graphs the eccentric distance sum is a novel topological index that offers a vast potential for structure activity property relationships for a graph g it is defined as d g v ve v d v where e v is the eccentricity of the vertex v and d v u v g d u v is the sum of all distances from the vertex v motivated by g yu l feng a ilic on the eccentric distance sum of trees and unicyclic graphs j math anal appl 375 2011 934 944 in this paper we characterize the extremal trees and graphs with maximal eccentric distance sum various lower and upper bounds for the eccentric distance sum in terms of other graph invariants including the wiener index the degree distance eccentric connectivity index independence number connectivity matching number chromatic number and clique number are established in addition we present explicit formulae for the values of eccentric distance sum for the cartesian product applied to some graphs of chemical interest like nanotubes and nanotori

connective steiner 3 eccentricity index and network similarity measure abstract for a set s v g in a network g the steiner distance dg s of s is the minimum size among all connected subnetworks whose vertex sets contain s the steiner k eccentricity k v of a vertex v of g is the maximum steiner distance among all k vertex set s which contains the vertex v i e e k v max d s s v g s k v s based on steiner k eccentricity the connective steiner k eccentricity index is introduced as a newly structural invariant some properties of the connective steiner 3 eccentricity index are investigated firstly we present an o n2 polynomial time algorithm to calculate the connective steiner 3 eccentricity index of trees secondly some optimal problems among some network classes are discussed as its application finally we consider the network similarity measure based on the connective steiner 3 eccentricity index by two different methods we study its advantages numerical results show that the measure based on the connective steiner 3 eccentricity index has more advantages than the ones based on other topological indices graph energy randic index the largest adjacent eigenvalue the largest laplacian eigenvalue

degeneracy in candecomp parafac and indscal explained for several three sliced arrays with a two valued typical rank the candecomp parafac cp method decomposes a three way array into a prespecified number r of rank 1 arrays by minimizing the sum of squares of the residual array the practical use of cp is sometimes complicated by the occurrence of so called degenerate sequences of solutions in which several rank 1 arrays become highly correlated in all three modes and some elements of the rank 1 arrays become arbitrarily large we consider the real valued cp decomposition of all known three sliced arrays i e of size p q 3 with a two valued typical rank these are the 5 3 3 and 8 4 3 arrays and the 3 3 4 and 3 3 5 arrays with symmetric 3 3 slices in the latter two cases cp is equivalent to the indscal model for a typical rank of m m 1 we consider the cp decomposition with r m of an array of rank m 1 we show that in most cases the cp objective function does not have a minimum but an infimum moreover any sequence of feasible cp solutions in which the objective value approaches the infimum will become degenerate we use the tools developed in stegeman 2006 who considers p p 2 arrays and present a framework of analysis which is of use to the future study of cp degeneracy related to a two valued typical rank moreover our examples show that cp uniqueness is not necessary for degenerate solutions to occur

eccentric distance sum a novel graph invariant for predicting biological and physical properties abstract eccentric distance sum a novel graph invariant with vast potential in structure activity property relationships has been conceptualized in the present study this graph invariant displayed high discriminating power with respect to both biological activity and physical properties the structure activity relationship of eccentric distance sum was investigated with regard to anti hiv activity of dihydroseselins the values of eccentric distance sum of each analogue in the data set were computed and active range identified subsequently biological activity was assigned to each analogue in the data set which was then compared with the reported anti hiv activity of dihydroseselin analogues surprisingly the accuracy of prediction was found to be more than 88 with regard to anti hiv activity on the other hand investigations pertaining to quantitative structure property relationship of the novel graph invariant with regard to various physical properties of diverse nature for data sets consisting of primary amines secondary amines and alcohols revealed correlation percentages ranging from 93 to 99 the over all results with regard to structure activity and quantitative structure property studies using eccentric distance sum were better than the corresponding values obtained using wiener s index

simplicity of core arrays in three way principal component analysis and the typical rank of p q 2 arrays abstract interpreting the solution of a principal component analysis of a three way array is greatly simplified when the core array has a large number of zero elements the possibility of achieving this has recently been explored by rotations to simplicity or to simple targets on the one hand and by mathematical analysis on the other in the present paper it is shown that a p q 2 array with p q 2 can almost surely be transformed to have all but 2 q elements zero it is also shown that arrays of that form have three way rank p at most this has direct implications for the typical rank of p q 2 arrays also when p q when p 2q the typical rank is 2 q when q it is p and when p q the rank is typically almost surely p or p 1 these typical rank results pertain to the decomposition of real valued three way arrays in terms of real valued rank one arrays and do not apply in the complex setting where the typical rank of p q 2 arrays is also min p 2q when p q but it is p when p q

some extremal results on the connective eccentricity index of graphs abstract the connective eccentricity index cei of a graph g is defined as ce g v i v g d v i e v i where e v i and d v i are the eccentricity and the degree of vertex v i respectively in g in this paper we obtain some lower and upper bounds on the connective eccentricity index for all trees of order n and with matching number and characterize the corresponding extremal trees and the maximal graphs of order n and with matching number and n edges have been determined which maximize the connective eccentricity index also the extremal graphs with maximal connective eccentricity index are completely characterized among all connected graphs of order n and with matching number moreover we establish some relations between connective eccentricity index and eccentric connectivity index as another eccentricity based invariant of graphs

application of graph theory relationship of eccentric connectivity index and wiener s index with anti inflammatory activity abstract graph theory was successfully applied in developing a relationship between chemical structure and biological activity the relationship of two graph invariants the eccentric connectivity index and the wiener s index was investigated with regard to anti inflammatory activity for a data set consisting of 76 pyrazole carboxylic acid hydrazide analogues the values of the eccentric connectivity index and the wiener s index of each analogue in the data set were computed and active ranges were identified subsequently each analogue was assigned a biological activity that was compared with the anti inflammatory activity reported as percent reduction in paw swelling prediction with an accuracy of 90 was obtained using the eccentric connectivity index as compared to 84 in the case of wiener s index

simplicity and typical rank results for three way arrays matrices can be diagonalized by singular vectors or when they are symmetric by eigenvectors pairs of square matrices often admit simultaneous diagonalization and always admit block wise simultaneous diagonalization generalizing these possibilities to more than two non square matrices leads to methods of simplifying three way arrays by nonsingular transformations such transformations have direct applications in tucker pca for three way arrays where transforming the core array to simplicity is allowed without loss of fit simplifying arrays also facilitates the study of array rank the typical rank of a three way array is the smallest number of rank one arrays that have the array as their sum when the array is generated by random sampling from a continuous distribution in some applications the core array of tucker pca is constrained to have a vast majority of zero elements both simplicity and typical rank results can be applied to distinguish constrained tucker pca models from tautologies an update of typical rank results over the real number field is given in the form of two tables

on the extremal properties of the average eccentricity the eccentricity of a vertex is the maximum distance from it to another vertex and the average eccentricity ecc g of a graph g is the mean value of eccentricities of all vertices of g the average eccentricity is deeply connected with a topological descriptor called the eccentric connectivity index defined as a sum of products of vertex degrees and eccentricities in this paper we analyze extremal properties of the average eccentricity introducing two graph transformations that increase or decrease ecc g furthermore we resolve four conjectures obtained by the system autographix about the average eccentricity and other graph parameters the clique number and the independence number refute one autographix conjecture about the average eccentricity and the minimum vertex degree and correct one autographix conjecture about the domination number

a short and unified proof of yu et al ʼs two results on the eccentric distance sum abstract the eccentric distance sum eds is a novel topological index that offers a vast potential for structure activity property relationships for a connected graph g the eccentric distance sum is defined as d g v v g ec g v d g v where ec g v is the eccentricity of a vertex v in g and d g v is the sum of distances of all vertices in g from v more recently yu et al g yu l feng a ilic on the eccentric distance sum of trees and unicyclic graphs j math anal appl 375 2011 99 107 proved that for an n vertex tree t d t 4 n 2 9 n 5 with equality holding if and only if t is the n vertex star s n and for an n vertex unicyclic graph g d g 4 n 2 9 n 1 with equality holding if and only if g is the graph obtained by adding an edge between two pendent vertices of n vertex star in this note we give a short and unified proof of the above two results

on the connective eccentricity index of trees and unicyclic graphs with given diameter abstract the connective eccentricity index cei of a graph g is defined as ce g v v g d v e v where e v d v denote the eccentricity and the degree of v respectively in this paper we investigate the connective eccentricity index of trees and unicyclic graphs on n vertices with diameter d we also characterize the extremal graphs

symmetry transformations for square sliced three way arrays with applications to their typical rank the typical 3 tensorial rank has been much studied over algebraically closed fields but very little has been achieved in the way of results pertaining to the real field the present paper examines the typical 3 tensorial rank over the real field when the slices of the array involved are square matrices the typical rank of 3 3 3 arrays is shown to be five the typical rank of p q q arrays is shown to be larger than q 1 unless there are only two slices p 2 or there are three slices of order 2 2 p 3 and q 2 the key result is that when the rank is q 1 there usually exists a rank preserving transformation of the array to one with symmetric slices

the eccentric connectivity index of nanotubes and nanotori let g be a molecular graph the eccentric connectivity index x c g is defined as x c g email protected u v g deg g u e g u where deg g u denotes the degree of vertex u and e g u is the largest distance between u and any other vertex v of g in this paper exact formulas for the eccentric connectivity index of tuc 4c 8 s nanotube and tc 4c 8 s nanotorus are given

typical ranks for m n m 1 n tensors with m n abstract in various application fields tensor type data are used recently and then a typical rank is important there may be more than one typical ranks over the real number field it is well known that the set of 2 n n tensors has two typical ranks n n 1 for n 2 that the set of 3 4 8 tensors has two typical ranks 8 9 and that the set of 4 4 12 tensors has two typical ranks 12 13 in this paper we show that the set of m n m 1 n tensors with m n has two typical ranks m 1 n m 1 n 1 if m n where is the hurwitz radon function defined as n 2 b 8 c for nonnegative integers a b c such that n 2 a 1 2 b 4 c and 0 b 4

generic and typical ranks of multi way arrays abstract the concept of tensor rank was introduced in the 20s in the 70s when methods of component analysis on arrays with more than two indices became popular tensor rank became a much studied topic the generic rank may be seen as an upper bound to the number of factors that are needed to construct a random tensor we explain in this paper how to obtain numerically in the complex field the generic rank of tensors of arbitrary dimensions based on terracini s lemma and compare it with the algebraic results already known in the real or complex fields in particular we examine the cases of symmetric tensors tensors with symmetric matrix slices complex tensors enjoying hermitian symmetries or merely tensors with free entries

a comparative analysis of new graph distance measures and graph edit distance the problem of determining the structural similarity or distance between graphs got considerable attention during the last decades consequently various similarity and distance measures for networks have already been investigated dehmer etal studied a new distance measure for graphs which is based on topological indices an interesting problem is to compare our class of measures with the well known graph edit distance which has been studied extensively and often serves as a benchmark measure but note that ged is generally np hard in this paper we compare the edit distance with our graph distance measures in order to do so we use some well known topological indices such as graph energy wiener index randi index and graph entropy by using several special graph classes numerical results reveal that the graph distance measure based on graph energy approximates ged well this fact could trigger an important research direction for studying graph edit distance and other comparative network measures