q analogs of distance matrices of 3 hypertrees abstract we consider the distance matrix of trees in 3 uniform hypergraphs which we call 3 hypertrees we give a formula for the inverse of a few q analogs of distance matrices of 3 hypertrees t some results are analogs of results by bapat et al for graphs we give an alternate proof of the result that the determinant of the distance matrix of a 3 hypertree t depends only on n the number of vertices of t further we give a pfaffian identity for a principal submatrix of some skew symmetrized distance matrices of 3 hypertrees when we fix an ordering of the vertices and assign signs appropriately a result of graham hoffman and hosoya relates the determinant of the distance matrix of a graph and the determinants of its two connected blocks when the graph has as blocks a fixed connected graph h which satisfy some conditions we give a formula for the inverse of its distance matrix this result generalises a result of graham and lovasz when each block of g is a fixed graph g we also give some corollaries about the sum of the entries of the inverse of the distance matrix of g and some of its analogs

a characterization of codes meeting the griesmer bound for any binary linear code of length n dimension k and minimum distance d the griesmer bound says that uif128 1 in this paper we completely characterize all codes which meet the griesmer bound with equality and for which d 2usuk 1 in particular we prove belov s conjecture

distance spectra of graphs a survey abstract in 1971 graham and pollack established a relationship between the number of negative eigenvalues of the distance matrix and the addressing problem in data communication systems they also proved that the determinant of the distance matrix of a tree is a function of the number of vertices only since then several mathematicians were interested in studying the spectral properties of the distance matrix of a connected graph computing the distance characteristic polynomial and its coefficients was the first research subject of interest thereafter the eigenvalues attracted much more attention in the present paper we report on the results related to the distance matrix of a graph and its spectral properties

a solution to an open problem for wave equations with generalized wentzell boundary conditions in this paper we solve an open problem put forward by a favini g r goldstein j a goldstein and s romanelli concerning the mixed problem for wave equations with generalized wentzell boundary conditions as a consequence we also develop the previous wellposedness result regarding the mixed problem for heat equations with generalized wentzell boundary conditions

on the 162 8 80 codes constructions of 162 8 80 and 159 8 78 codes are given this solves the open problems of finding the minimum length of binary codes of dimension s and minimum distances 78 and 80 respectively

identities for minors of the laplacian resistance and distance matrices it is shown that if l and d are the laplacian and the distance matrix of a tree respectively then any minor of the laplacian equals the sum of the cofactors of the complementary submatrix of d up to sign and a power of 2 an analogous more general result is proved for the laplacian and the resistance matrix of any graph a similar identity is proved for graphs in which each block is a complete graph on r vertices and for q analogues of such matrices of a tree our main tool is an identity for the minors of a matrix and its inverse

the change of distance energy of some special complete multipartite graphs due to edge deletion abstract let k p 1 p 2 p r be a complete r partite graph with r 2 and p i 2 varghese et al 2018 14 conjectured that the distance energy of k p 1 p 2 p r is always increased when an edge is deleted in this paper we prove that it is true for k p p p and the tripartite turan graph t n 3 subsequently it is proved that for any positive integer r there exists a class of graphs that have r positive eigenvalues but the distance energy of these graphs is always increased when an edge is deleted

some new results for optimal ternary linear codes let d sub 3 n k be the maximum possible minimum hamming distance of a ternary n k d code for given values of n and k we describe a package for code extension and use this to prove some new exact values of d sub 3 n k moreover we classify the ternary n k d sub 3 n k codes for some values of n and k

an almost full embedding of the category of graphs into the category of groups abstract we construct a functor f g raphs g roups which is faithful and almost full in the sense that every nontrivial group homomorphism f x f y is a composition of an inner automorphism of fy and a homomorphism of the form ff for a unique map of graphs f x y when f is composed with the eilenberg mac lane space construction k f x 1 we obtain an embedding of the category of graphs into the unpointed homotopy category which is full up to null homotopic maps we provide several applications of this construction to localizations i e idempotent functors we show that the questions 1 is every orthogonality class reflective 2 is every orthogonality class a small orthogonality class have the same answers in the category of groups as in the category of graphs in other words they depend on set theory 1 is equivalent to weak vop nka s principle and 2 to vop nka s principle additionally the second question considered in the homotopy category is also equivalent to vop nka s principle

on distance matrices and laplacians we consider distance matrices of certain graphs and of points chosen in a rectangular grid formulae for the inverse and the determinant of the distance matrix of a weighted tree are obtained results concerning the inertia and the determinant of the distance matrix of an unweighted unicyclic graph are proved if d is the distance matrix of a tree then we obtain certain results for a perturbation of d 1 as an example it is shown that if l is the laplacian matrix of an arbitrary connected graph then d 1 l 1 is an entrywise positive matrix we consider the distance matrix of a subset of a rectangular grid of points in the plane if we choose m k 1 points not containing a closed path in an m k grid then a formula for the determinant of the distance matrix of such points is obtained

the distance matrix of a tree with weights on its arcs abstract an ordered pair of two adjacent vertices in a graph is called an arc in this paper we give the determinant and the inverse whenever it exists of the distance matrix of an arc weighted tree with arc weights in a ring these generalize known results we also give examples that cannot be dealt with by known results however our results can be applied for these examples

new binary linear codes which are dual transforms of good codes if c is a binary linear code one may choose a subset s of c and form a new code c sub s sup t which is the row space of the matrix having the elements of s as its columns one way of picking s is to choose a subgroup h of aut c and let s be some h stable subset of c using primarily this method for picking s we obtain codes with record breaking parameters 92 8 44 105 8 50 108 8 52 79 9 36 123 9 58 130 9 62 175 9 84 242 9 118 and 245 9 120

inverse of the distance matrix of a weighted cactoid digraph abstract a weighted cactoid digraph is a strongly connected directed graph whose blocks are weighted directed cycles such that any two distinct weighted directed cycles of this graph share at most one common vertex in this paper we give the determinant and the inverse of the distance matrix of a weighted cactoid digraph which imply graham and pollak s formula and the inverse of the distance matrix of a tree

generation of localization in a discrete chain with periodic boundary conditions numerical and analytical results we study the generation of localization in a discrete chain composed of n subsystems and periodic boundary conditions strongly localized motions are studied that is time periodic motions where nearly all of the energy is spatially confined to a single subsystem for varying n numerical results indicate that the strongly localized solutions are generated through a bifurcation from an in phase spatially extended solution however in the limit as n rightarrow infty the bifurcation point tends to infinity and a smooth transition from localization to nonlocalization occurs we then present an analytic technique to complement the numerical results it is based on the matching local asymptotic expansions of a solution branch using pade approximants this leads to global analytic representations of the considered solutions valid over the entire range of the control parameter

someinverseproblemsforellipticmatriceswith zero diagonal abstract we present explicit formulae which allow us to construct elliptic matrices with zero diagonal from a given spectrum as well as from the complete interlacing system

the inertia and energy of distance matrices of complete k partite graphs abstract for a distance matrix d g its inertia is the triple of integers n d n 0 d n d where n d n 0 d n d denote the number of positive 0 negative eigenvalues of d g respectively the d energy is the sum of the absolute eigenvalues of d g in this paper we first study the inertia of distance matrices of complete k partite graphs then as applications we not only prove a conjecture proposed by caporossi et al 2009 in 5 in a different way from stevanovic et al 2013 11 but also obtain the formula of the d energy of the remaining complete k partite graphs at last we obtain the graphs with the maximum resp minimum d energy among all the complete k partite graphs with n vertices

the nonexistence of certain binary linear codes some of the upper bounds given by t verhoeff 1987 are improved by proving the nonexistence of codes with certain parameters necessary preliminary results are stated the results give rise to upper bounds on d n k for many further values of n k by using simple standard techniques the further bounds would automatically follow from theorems by using verhoeff s computerized updating program but the authors state them here because several of them are required in the proofs of subsequent theorems

a de giorgi nash type theorem for time fractional diffusion equations we study the regularity of weak solutions to linear time fractional diffusion equations in divergence form of arbitrary time order alpha in 0 1 the coefficients are merely assumed to be bounded and measurable and they satisfy a uniform parabolicity condition our main result is a de giorgi nash type theorem which gives an interior holder estimate for bounded weak solutions in terms of the data and the l infty bound of the solution the proof relies on new a priori estimates for time fractional problems and uses de giorgi s technique and the method of non local growth lemmas which has been introduced recently in the context of nonlocal elliptic equations involving operators like the fractional laplacian

the coefficients of the immanantal polynomial abstract an expression of the coefficient of immanantal polynomial of an n n matrix is present moreover we give expressions of the coefficient of immanantal polynomials of combinatorial matrices adjacency matrix laplacian matrix signless laplacian matrix as applications we show that the immanantal polynomials for laplacian matrix and signless laplacian matrix of bipartite graphs are the same this is a generalization of the characteristic polynomial for laplacian matrix and signless laplacian matrix of bipartite graphs furthermore we consider the relations between the characteristic polynomial and the immanantal polynomial for trees

laplacian immanantal polynomials and the gts poset on trees abstract let t be a tree on n vertices with laplacian l t and let gts n be the generalised tree shift poset on the set of unlabelled trees on n vertices inequalities are known for coefficients of the characteristic polynomial of l t as we go up the poset gts n in this work we generalise these inequalities to the q laplacian l t q of t and to the coefficients of all immanantal polynomials

the inertia and energy of the distance matrix of a connected graph abstract let g be a connected graph and d g be the distance matrix of g suppose that 1 d 2 d n d are the d eigenvalues of g in this paper we show that the distance matrix of a clique tree is non singular moreover we also prove that the distance matrix of a clique tree has exactly one positive d eigenvalue in addition we determine the extremal graphs with maximum and minimum distance energy among all clique trees

on the distance spectral radius of bipartite graphs abstract in this paper we determine the unique graph with minimum distance spectral radius among all connected bipartite graphs of order n with a given matching number moreover we characterize the graphs with minimal distance spectral radius in the class of all connected bipartite graphs with a given vertex connectivity

some results for linear binary codes with minimum distance 5 and 6 we prove that a linear binary code with parameters 34 24 5 does not exist also we characterize some codes with minimum distance 5 and 6

on optimal nonlinear systematic codes most bounds on the size of codes hold for any code whether linear or not notably the griesmer bound holds only in the linear case and so optimal linear codes are not necessarily optimal codes in this paper we identify code parameters q d k namely field size minimum distance and combinatorial dimension for which the griesmer bound also holds in the systematic nonlinear case moreover we show that the griesmer bound does not necessarily hold for a systematic code by explicit construction of a family of optimal systematic binary codes on the other hand we are able to provide some versions of the griesmer bound holding for all the systematic codes

spherically punctured biorthogonal codes consider a binary reed muller code rm r m defined on the hypercube bb f2m and let all code positions be restricted to the m tuples of a given hamming weight b in this paper we specify this single layer construction obtained from the biorthogonal codes rm 1 m and the hadamard codes h m both punctured codes inherit some recursive properties of the original rm codes however they cannot be formed by the recursive plotkin construction we first observe that any code vector in these codes has hamming weight defined by the weight w of its information block more specifically this weight depends on the absolute values of the krawtchouk polynomials kbm w we then study the properties of the krawtchouk polynomials and show that the minimum code weight of a single layer code rm 1 m b is achieved at the minimum input weight w 1 for any we further refine our codes by limiting the possible weights w of the input information blocks as a result some of the designed code sequences meet or closely approach the griesmer bound finally we consider more general punctured codes whose positions form several spherical layers

using combinatorial optimization to design good unit memory convolutional codes a method for designing good unit memory convolutional codes is presented the method is based on the decomposition of the original problem into two easier subproblems that can be formulated as optimization problems and solved by efficient heuristic search algorithms the efficacy of this method is demonstrated by a table containing 33 new unit memory convolutional codes n k with 5

two laplacians for the distance matrix of a graph abstract we introduce a laplacian and a signless laplacian for the distance matrix of a connected graph called the distance laplacian and distance signless laplacian respectively we show the equivalence between the distance signless laplacian distance laplacian and the distance spectra for the class of transmission regular graphs there is also an equivalence between the laplacian spectrum and the distance laplacian spectrum of any connected graph of diameter 2 similarities between n as a distance laplacian eigenvalue and the algebraic connectivity are established

the linear programming bound for binary linear codes combining delsarte s 1973 linear programming bound with the information that certain weights cannot occur new upper bounds for d sub min n k the maximum possible minimum distance of a binary linear code with given word length n and dimension k are derived

macwilliams identities and coordinate partitions abstract any partition of the coordinate set of a linear code is shown to correspond to a set of generalized macwilliams identities thus a well chosen partition yields a promising method to settle existence and uniqueness problems a short proof of an extension of the assmus mattson theorem is given in the nonlinear case a generalization of the delsarte inequalities is obtained

some new results on the minimum length of binary linear codes of dimension nine let n k d be the smallest integer n for which a binary linear code of length n dimension k and minimum distance d exists using the residual code technique the macwilliams identities and the weight distribution of appropriate reed muller codes we prove that n 9 64 133 n 9 120 spl ges 244 n 9 124 252 and n 9 184 371 we also show that puncturing a known 322 9 160 code yields length optimal codes with parameters 319 9 158 315 9 156 and 312 9 154

on a 145 32 binary cyclic code we discovered that a 145 32 binary cyclic code generated by the product of an irreducible polynomial of degree 28 with exponent 145 and the polynomial x sup 4 x sup 3 x sup 2 x 1 has a minimum distance of 44

modular invariant of quantum tori the quantum modular invariant j qt of 2r is defined as a discontin uous pgl2 z invariant multi valued map using the distance to the nearest integer function k k for 2 q it is shown that j qt 1 and for quadratic irrationalities pari gp experiments suggest that j qt is a finite set in the case of the golden mean we produce explicit formulas involving weighted versions of the rogers ramanujan functions for the experimental supremum and infimum we then define a universal modular invariant j a mod c as a continuous and single valued map of ultrasolenoids such that 1 the classical modular invariant is a quotient of the re striction of j to a subsolenoid mod cl a mod fibering over the classical moduli space of elliptic curves and 2 the quantum modular invariant is a quotient of the restric tion of j to a subsolenoid mod qt a mod fibering over the moduli space of elliptic curves equipped with a kronecker foliation

on the gap of finite metric spaces of p negative type let x d be a metric space of p negative type recently i doust and a weston introduced a quantification of the p negative type property the so called gap of x this paper gives some formulas for the gap of a finite metric space of strict p negative type and applies them to evaluate for some concrete finite metric spaces

optimum codes of dimension 3 and 4 over gf 4 n sub q k d the length of a q ary optimum code for given k and d for q 4 and k 3 4 is discussed the problem is completely solved for k 3 and the exact value of n sub 4 4 d is determined for all but 52 values of d

new combinatorial interpretations of ramanujan s partition congruences mod 5 7 and 11 soit p n le nombre de partitions non restreintes de n on donne de nouvelles interpretations combinatoires des congruences p 5n 4 p 7n 5 et p 11n 6 0 mod 5 7 et 11 respectivement

new bounds on binary linear codes of dimension eight corresp let n k d be the smallest integer n such that a binary linear code of length n dimension k and minimum distance at least d exists new results are given that improve the best previously known bounds on n 8 d

distance energy change of complete bipartite graph due to edge deletion abstract the distance matrix distance eigenvalue and distance energy of a connected graph have been studied intensively in the literature we propose a new problem of studying how the distance energy changes when an edge is deleted in this paper we prove that the distance energy of a complete bipartite graph is always increased when an edge is deleted

some inequalities for the q beta and the q gamma functions via some q integral inequalities some new inequalities for the q gamma the q beta and the q analogue of the psi functions are established via some q integral inequalities

there is no binary 25 8 10 code the existence of a binary 25 8 10 code is considered it is shown that such a code must have a generator matrix of a specific form however all generator matrices of this form were tested and none generated a 25 8 10 code thus such a code does not exist

new optimal binary linear codes of dimensions 9 and 10 eighteen new codes are presented which improve the bounds on maximum minimum distance for binary linear codes they are rate m r pm r spl ges 1 degenerate quasi cyclic qc codes based on the known upper bounds six of these new codes are optimal in addition five two weight qc codes of dimension 8 are given

an improved sectional anosov closing lemma we prove that all nonwandering points of a sectional anosov flow on a compact 3 manifold can be approximated by periodic points or by points for which the omega limit set is a singularity this improves the closing lemma in morales mich math j 56 1 29 53 2008 we also describe a sectional anosov flow for which the recurrent points are not dense in the nonwandering set

hook immanantal and hadamard inequalities for q laplacians of trees abstract let t be a tree on n vertices with laplacian matrix l and q laplacian l q let k be the character of the irreducible representation of s n indexed by the hook partition k 1 n k and let d k l be the normalized hook immanant of l corresponding to the character k inequalities for d k l as k increases are known by assigning a statistic to vertex orientations on trees we generalize these inequalities to the q analogue l q of l for all q r and to the bivariate q t laplacian l q t for some values q t our statistic based approach also generalizes several other inequalities including the changing index k l of the hadamard inequality for l to the matrix l q and l q t thus we extend several results about l to l q which includes the case when l q is not positive semidefinite

results on binary linear codes with minimum distance 8 and 10 all linear binary codes with minimum distance 8 and codimension up to 14 and all codes with minimum distance 10 and codimension up to 18 are classified nonexistence of codes with parameters 33 18 8 and 33 14 10 is proved this leads to 8 new exact bounds for binary linear codes primarily two algorithms considering the dual codes are used namely extension of dual codes with a proper coordinate and a fast algorithm for finding a maximum clique in a graph which is modified to find a maximum set of vectors with the right dependency structure

inverse of the distance matrix of a cactoid digraph abstract a strongly connected directed graph in which any two directed cycles of it share at most one common vertex is called a cactoid digraph let d be its distance matrix by a theorem of graham hoffman and hosoya we have det d 0 in this paper we give the formulas for both the determinant and the inverse of the distance matrix d for a cactoid digraph

on cartesian product of euclidean distance matrices abstract if a r m m and b r n n we define the product a b as a b a j n j m b where denotes the kronecker product and j n is the n n matrix of all ones we refer to this product as the cartesian product of a and b since if d 1 and d 2 are the distance matrices of graphs g 1 and g 2 respectively then d 1 d 2 is the distance matrix of the cartesian product g 1 g 2 we study cartesian products of euclidean distance matrices edms we prove that if a and b are edms then so is the product a b we show that if a is an edm and u is symmetric then a u is an edm if and only if u c j n for some c it is shown that for edms a and b a b is spherical if and only if both a and b are spherical if a and b are edms then we derive expressions for the rank and the moore penrose inverse of a b in the final section we consider the product a b for arbitrary matrices for a r m m b r n n we show that all nonzero minors of a b of order m n 1 are equal an explicit formula for a nonzero minor of order m n 1 is proved the result is shown to generalize the familiar fact that the determinant of the distance matrix of a tree on n vertices does not depend on the tree and is a function only of n

distance spectral radius of graphs with r pendent vertices abstract let g n r be the class of all connected graphs of order n with r pendent vertices in this paper we determine the unique graph with minimal distance spectral radius in g n r in addition we determine the unique graph with maximal distance spectral radius in g n r for each r 2 3 n 3 n 2 n 1

extremal cacti of given matching number with respect to the distance spectral radius a cactus is a connected graph in which any two cycles have at most one common vertex the distance spectral radius g of a graph g is the largest eigenvalue of the distance matrix d g recently many researchers proposed the use of g as a molecular structure descriptor of alkanes in this paper we characterize n vertex cyclic cactus with given matching number m which minimizes the distance spectral radius the resulting cactus also minimizes the hosoya index the wiener index and the randic index in the same class of graphs

proof of a conjecture involving the second largest d eigenvalue and the number of triangles abstract let g be a connected graph of order n with tr triangles and d be the distance matrix of g let 1 d 2 d n d be the d eigenvalue of the graph g fajtlowicz 1998 4 conjectured that 2 d tr when the independent number g 2 in this paper the conjecture is confirmed and the extremal graph when the equality holds is characterized

the product distance matrix of a tree with matrix weights on its arcs abstract let t be a tree with vertex set n 1 2 n for each i n let m i be a positive integer an ordered pair of two adjacent vertices is called an arc each arc i j of t has a weight w i j which is an m i m j matrix for two vertices i j n let the unique directed path from i to j be p i j x 0 x 1 x d where d 1 x 0 i and x d j define the product distance from i to j to be the m i m j matrix m i j w x 0 x 1 w x 1 x 2 w x d 1 x d let n i 1 n m i the n n product distance matrix d of t is a partitioned matrix whose i j block is the matrix m i j we give a formula for det d when det d 0 the inverse of d is also obtained these generalize known results for the product distance matrix when either the weights are real numbers or m 1 m 2 m n s and the weights w i j w j i w e for each edge e i j e t

distance matrix and laplacian of a tree with attached graphs a tree with attached graphs is a tree together with graphs defined on its partite sets we introduce the notion of incidence matrix laplacian and distance matrix for a tree with attached graphs formulas are obtained for the minors of the incidence matrix and the laplacian and for the inverse and the determinant of the distance matrix the case when the attached graphs themselves are trees is studied more closely several known results including the matrix tree theorem are special cases when the tree is a star the case when the attached graphs are paths is also of interest since it is related to the transportation problem

product distance matrix of a tree with matrix weights abstract let t be a tree on n vertices and let the n 1 edges e 1 e 2 e n 1 have weights that are s s matrices w 1 w 2 w n 1 respectively for two vertices i j let the unique ordered path between i and j be p i j e r 1 e r 2 e r k define the distance between i and j as the s s matrix e i j p 1 k w e p consider the n s n s matrix d whose i j th block is the matrix e i j we give a formula for det d and for its inverse when it exists these generalize known results for the product distance matrix when the weights are real numbers

further classifications of codes meeting the griesmer bound corresp for any n k d binary linear code the griesmer bound says that n geq sum i 0 k 1 lceil d 2 i rceil where lceil x rceil denotes the smallest integer geq x we consider codes meeting the griesmer bound with equality these codes have parameters left s 2 k 1 sum i 1 p 2 u i 1 k s2 k 1 sum i 1 p 2 u i 1 right where k u 1 cdots u p geq 1 we characterize all such codes when p 2 or u i 1 u i geq 2 for 2 leq i leq p

the inverse of the distance matrix of a distance well defined graph abstract a square matrix l is called a laplacian like matrix if l j 0 and j t l 0 a square matrix d is left or right laplacian expressible if there exist a number 0 a column vector satisfying t j 1 and a square matrix l such that t d j t l d i j t and l j 0 or d j d l i j t and j t l 0 we consider the generalized distance matrix d see definition 4 1 of a graph whose blocks correspond to left or right laplacian expressible matrices then d is also left or right laplacian expressible and the inverse d 1 when it exists can be expressed as the sum of a laplacian like matrix and a rank one matrix

new code parameters from reed solomon subfield codes we determine the dimensions of subfield codes of reed solomon codes and construct certain extensions and lengthenings of these codes we start from the duals using the language of orthogonal arrays as a first result this allows us to obtain a fair number of improvements in the list of binary ternary and quaternary linear codes with largest known minimal distance

inverse of the distance matrix of a cycle clique graph abstract a connected graph g all of whose blocks are cycles or cliques is called a cycle clique graph let d be the distance matrix of g by a theorem of graham et al we have det d 0 if all cycle blocks have odd vertices in this paper we give the formula for the inverse of d

the smallest length of eight dimensional binary linear codes with prescribed minimum distance let n 8 d be the smallest integer n for which a binary linear code of length n dimension 8 and minimum distance d exists we prove that n 8 18 42 n 8 26 58 n 8 28 61 n 8 30 65 n 8 34 74 n 8 36 77 n 8 38 81 n 8 42 89 and n 8 60 124 after these results all values of n 8 d are known

resistance matrices of graphs with matrix weights abstract the resistance matrix of a simple connected graph g is denoted by r and is defined by r r i j where r i j is the resistance distance between the vertices i and j of g in this paper we consider the resistance matrix of weighted graph with edge weights being positive definite matrices of same size we derive a formula for the determinant and the inverse of the resistance matrix then we establish an interlacing inequality for the eigenvalues of resistance and laplacian matrices of tree using this interlacing inequality we obtain the inertia of the resistance matrix of tree

inverses of q distance matrices of a tree the determinant and the inverse of the distance matrix of a tree have been investigated in the literature following the classical formulas due to graham and pollak for the determinant and due to graham and lovasz for the inverse we consider two q analogs of the distance matrix of a tree and obtain formulas for the inverses of the two distance matrices yan and yeh have previously obtained expressions for the determinants of the two distance matrices some related results are proved

distance spectra and distance energy of integral circulant graphs the distance energy of a graph g is a recently developed energy type invariant defined as the sum of absolute values of the eigenvalues of the distance matrix of g there was a vast research for the pairs and families of non cospectral graphs having equal distance energy and most of these constructions were based on the join of graphs a graph is called circulant if it is cayley graph on the circulant group i e its adjacency matrix is circulant a graph is called integral if all eigenvalues of its adjacency matrix are integers integral circulant graphs play an important role in modeling quantum spin networks supporting the perfect state transfer in this paper we characterize the distance spectra of integral circulant graphs and prove that these graphs have integral eigenvalues of distance matrix d furthermore we calculate the distance spectra and distance energy of unitary cayley graphs in conclusion we present two families of pairs g1 g2 of integral circulant graphs with equal distance energy in the first family g1 is subgraph of g2 while in the second family the diameter of both graphs is three

an updated table of minimum distance bounds for binary linear codes in 1973 helgert and sfinaff published a table of upper and lower bounds on the maximum minimum distance for binary linear error correcting codes up to length 127 this article presents an updated table incorporating numerous improvements that have appeared since then to simplify the updating task the author has developed a computer program that systematically investigates the consequences of each improvement by applying several well known general code construction techniques this program also made it possible to check the original table furthermore it offers a quick and reliable update service for future improvements

squared distance matrix of a tree inverse and inertia abstract let t be a tree with vertices v t 1 n the distance between vertices i j v t denoted d i j is defined to be the length the number of edges of the path from i to j we set d i i 0 i 1 n the squared distance matrix of t is the n n matrix with i j element equal to 0 if i j and d i j 2 if i j it is known that is nonsingular if and only if the tree has at most one vertex of degree 2 we obtain a formula for 1 if it exists when the tree has no vertex of degree 2 the formula is particularly simple and depends on a certain two step laplacian of the tree we determine the inertia of the inverse and the inertia of the edge orientation matrix are also described

a nonterminating 8 φ 7 summation for the root system c r using multiple q integrals and a determinant evaluation we establish a nonterminating 8 7 summation for the root system cr we also give some important specializations explicitly

double centralizing theorems for the alternating groups let v n be the n fold tensor product of a vector space v following i schur we consider the action of the symmetric group sn on v n by permuting coordinates in the super z2 graded case v v0 v1 a sign is added these actions give rise to the corresponding schur algebras s sn v here s sn v is compared with s an v the schur algebra corresponding to the alternating subgroup an sn while in the classical signless case these two schur algebras are the same for n large enough it is proved that in the super case where dim v0 dim v1 s an v is isomorphic to the cross product algebra s an v s sn v z2

large e rings exist a ring r with 1 is called an e ring provided hom r r z r under the map cp 1 the class of e rings was defined and studied by schultz s in 1973 and further investigated by bowshell and schultz bs in 1977 examples of e rings include z n subrings of q and pure subrings of the ring of p adic integers more interesting examples are the torsion free e rings of finite rank these were characterized in bs as those rings quasi isomorphic to r x r x x r where each ri is a strongly indecomposable subring of an algebraic number field and hom r rj 0 for i j in spite of their seemingly specialized nature such rings have played an important role in the theory of torsion free groups of finite rank dating back to beaumont pierce bpl bp2 and pierce p in 1960 1961 see also aprvw nr pv ri relatively little has been published on infinite rank torsion free e rings indeed until recently the only examples of these were provided by the pure subrings of the p adic integers in this paper the black box of shelah is used to construct a host of large e rings we show that any torsion free p reduced p cotorsion free commutative ring s may be embed88 0021 8693 87 3 00

on euclidean distance matrices abstract if a is a real symmetric matrix and p is an orthogonal projection onto a hyperplane then we derive a formula for the moore penrose inverse of pap as an application we obtain a formula for the moore penrose inverse of an euclidean distance matrix edm which generalizes formulae for the inverse of a edm in the literature to an invertible spherical edm we associate a laplacian matrix which we define as a positive semidefinite n n matrix of rank n 1 and with zero row sums and prove some properties known results for distance matrices of trees are derived as special cases in particular we obtain a formula due to graham and lovasz for the inverse of the distance matrix of a tree it is shown that if d is a nonsingular edm and l is the associated laplacian then d 1 l is nonsingular and has a nonnegative inverse finally infinitely divisible matrices are constructed using edms

on the covering radius of binary linear codes meeting the griesmer bound let g k d sum i 0 k 1 lceil d 2 i rceil by the griesmer bound n geq g k d for any binary linear n k d code let s lceil d 2 k 1 rceil then s can be interpreted as the maximum number of occurrences of a column in the generator matrix of any code with parameters g k d k d let rho be the covering radius of a g k d k d code it will be shown that rho leq d lceil s 2 rceil moreover the existence of a g k d k d code with rho d lceil s 2 rceil is equivalent to the existence of a g k 1 d k 1 d code for s leq 2 all g k d k d codes with rho d lceil s 2 rceil are described while for s 2 a sufficient condition for their existence is formulated

new constructions of codes meeting the griesmer bound for any binary linear n k d code the grfesmer bound says that n geq sum i 0 k 1 lceil d 2 i rceil we investigate codes that meet this bound with equality we give new descriptions of sum i 0 k 1 lceil d 2 i rceil k d codes that have earlier been constructed by solomon and stiffler belov and helleseth and van tilborg finally we show how to construct several new families of such codes with parameters not obtainable by any previous known constructions

graphs whose distance matrix has at most three negative eigenvalues abstract let d g be the distance matrix of a connected simple graph g the negative inertia of d g denoted by n d g is the number of negative eigenvalues of d g in this paper we determine all connected graphs g whose distance matrix d g has at most three negative eigenvalues

determinant of the distance matrix of a tree with matrix weights let t be a tree with n vertices and let d be the distance matrix of t according to a classical result due to graham and pollack the determinant of d is a function of n but does not depend on t we allow the edges of t to carry weights which are square matrices of a fixed order the distance matrix d of t is then defined in a natural way we obtain a formula for the determinant of d which involves only the determinants of the sum and the product of the weight matrices

a new class of codes meeting the griesmer bound an infinite sequence of k dimensional binary linear block codes is constructed with parameters n 2 k 2 k 2 15 d 2 k 1 2 k 3 8 k geq 7 for k geq 8 these codes are unique while there are five nonisomorphic codes for k 7 by shortening these codes in an appropriate way one finds codes meeting the griesmer bound for 2 k 1 2 k 3 15 leq d leq 2 k 1 2 k 3 8 k geq 7

new minimum distance bounds for certain binary linear codes let an n k d code denote a binary linear code of length n dimension k and minimum distance at least d define d n k as the maximum value of d for which there exists a binary linear n k d code t verhoeff 1989 has provided an updated table of bounds on d n k for 1

localizations of torsion free abelian groups ii abstract a homomorphism a b between abelian groups a b is called a localization of a if for each hom a b there is a unique end b such that it is well known that if a z then b is an e ring and 1 is the identity of b we investigate localizations of rank 1 groups a l q of type it turns out that localizations of l can be surprisingly complicated if l m is a localization and l is a subring of q then m is simply an e ring that is also an l module if l is not a subring things get more complicated if m m then tensor products of l and e rings come into play it is possible that m m and we can say very little in this case another topic under consideration are localizations of e rings frequently localizations of e rings are e rings again but we find examples where this is not the case

the x ray transform and its inversion for the series expansion basis functions in three dimensional tomography a formula for the three dimensional x ray transform and an expression for its inversion have been derived the spherical harmonics and the rotation matrices in terms of euler angles are used in the generalized fourier series expansion some properties of the transformed fourier components are discussed

decomposition of linear automata over residue rings into shift registers abstract a linear automaton of a over a residue ring n n e is in general not decomposable into a parallel connection of shift registers we are able to formulate necessary and sufficient conditions for such a decomposition by using the theory of artin local rings r and the r x module structure of a

eigenvalue monotonicity of q laplacians of trees along a poset abstract let t be a tree on n vertices with q laplacian matrix l t q let gts n be the generalized tree shift poset on the set of unlabelled trees with n vertices we prove that for all q r going up on gts n has the following effect the spectral radius of l t q increases while the smallest eigenvalue of l t q decreases we also prove that for all q r with q 1 going up on gts n increases the second smallest eigenvalue of l t q these generalize known results for eigenvalues of the laplacian matrix of trees as a corollary we obtain consequences about the eigenvalues of q t laplacians and exponential distance matrices of trees

a q analogue of the distance matrix of a tree we consider a q analogue of the distance matrix called the q distance matrix of an unweighted tree and give formulae for the inverse and the determinant which generalize the existing formulae for the distance matrix we obtain the smith normal form of the q distance matrix of a tree the relationship of the q distance matrix with the laplacian matrix leads to q analogue of the laplacian matrix of a tree some of whose properties are also given we study another matrix related to the distance matrix the exponential distance matrix and show its relationship with the q laplacian and the q distance matrix a formula for the determinant of the q distance matrix of a weighted tree is also given

characterization of symmetric m matrices as resistive inverses abstract we aim here at characterizing those nonnegative matrices whose inverse is an irreducible stieltjes matrix specifically we prove that any irreducible stieltjes matrix is a resistive inverse to do this we consider the network defined by the off diagonal entries of the matrix and we identify the matrix with a positive definite schrodinger operator whose ground state is determined by the lowest eigenvalue of the matrix and the corresponding positive eigenvector we also analyze the case in which the operator is positive semidefinite which corresponds to the study of singular irreducible symmetric m matrices