evaluation of the weight distribution of a class of cyclic codes based on index 2 gauss sums the duals of cyclic codes with two zeros have been extensively studied and their weight distributions have recently been evaluated in some cases in this paper we determine the weight distribution of a certain new class of such codes by computations involving index 2 gauss sums

novel inequalities for generalized graph entropies graph energies and topological indices the entropy of a graph is an information theoretic quantity for measuring the complexity of a graph after shannon introduced the entropy to information and communication many generalizations of the entropy measure have been proposed such as renyi entropy and daroczy entropy in this article we prove accurate connections inequalities between generalized graph entropies graph energies and topological indices additionally we obtain some extremal properties of nine generalized graph entropies by employing graph energies and topological indices

trees with minimal laplacian coefficients let g be a simple undirected graph with the characteristic polynomial of its laplacian matrix l g p g m email protected k 0 n 1 kc email protected 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 the graph in this paper we characterize n vertex trees with given matching number m which simultaneously minimize all laplacian coefficients the extremal tree a n m is a spur obtained from the star graph s n m 1 with n m 1 vertices by attaching a pendant edge to each of certain m 1 non central vertices of s n m 1 in particular a n m minimizes the wiener index the modified hyper wiener index and the recently introduced incidence energy of trees defined as ie g email protected k 0 email protected k where m k are the eigenvalues of signless laplacian matrix q g d g a g we introduced a general r transformation which decreases all laplacian coefficients simultaneously in conclusion we illustrate on examples of wiener index and incidence energy that the opposite problem of simultaneously maximizing all laplacian coefficients has no solution

analyzing lattice networks through substructures abstract analyzing the topology of network structures is an important topic studied from many different aspects of science and mathematics the wiener polarity index number of unordered pairs of vertices at distance 3 from each other is one of the representative descriptors of graph structures it was computed for several lattice networks by chen et al 11 in an effort to understand the properties of these networks the wiener polarity index is a variation of the classic distance based graph invariant the wiener index sum of distances between all pairs of vertices which is known to be closely related to the number of substructures in this paper we examine the numbers of various subgraphs of order 4 for these lattice networks in addition to confirming their symmetric nature comparing the numbers of various substructures leads to insights on other less trivial characteristics of these network structures of common interest

extremal energies of integral circulant graphs via multiplicativity abstract the energy of a graph is the sum of the moduli of the eigenvalues of its adjacency matrix integral circulant graphs can be characterised by their order n and a set d of positive divisors of n in such a way that they have vertex set z n z and edge set a b a b z n z gcd a b n d among integral circulant graphs of fixed prime power order p s those having minimal energy e min p s or maximal energy e max p s respectively are known we study the energy of integral circulant graphs of arbitrary order n with so called multiplicative divisor sets this leads to good bounds for e min n and e max n as well as conjectures concerning the true value of e min n

on the laplacian coefficients of tricyclic graphs abstract let g d e t i n l g k 0 n 1 k c k n k be the characteristic polynomial of the laplacian matrix of a graph g of order n in this paper we show that among all connected tricyclic graphs of order n the k th coefficient c k is smallest when the graph is b n 7 1 3 3 3 obtained from the complete graph k 4 by adding n 4 pendent vertices attached to the vertex of degree 3 and for some lemmas in c x he h y shan on the laplacian coefficients of bicyclic graphs discrete math 310 2010 3404 3412 we present a new method to prove them

nonbinary stabilizer codes over finite fields one formidable difficulty in quantum communication and computation is to protect information carrying quantum states against undesired interactions with the environment to address this difficulty many good quantum error correcting codes have been derived as binary stabilizer codes fault tolerant quantum computation prompted the study of nonbinary quantum codes but the theory of such codes is not as advanced as that of binary quantum codes this paper describes the basic theory of stabilizer codes over finite fields the relation between stabilizer codes and general quantum codes is clarified by introducing a galois theory for these objects a characterization of nonbinary stabilizer codes over fq in terms of classical codes over fq 2 is provided that generalizes the well known notion of additive codes over f4 of the binary case this paper also derives lower and upper bounds on the minimum distance of stabilizer codes gives several code constructions and derives numerous families of stabilizer codes including quantum hamming codes quadratic residue codes quantum melas codes quantum bose chaudhuri hocquenghem bch codes and quantum character codes the puncturing theory by rains is generalized to additive codes that are not necessarily pure bounds on the maximal length of maximum distance separable stabilizer codes are given a discussion of open problems concludes this paper

the laplacian incidence energy of graphs abstract the laplacian incidence energy of a graph is defined as the sum of the singular values of its normalized oriented incidence matrix in this paper we give sharp upper and lower bounds as well as the coulson integral formula for the laplacian incidence energy moreover we show a close relation of the laplacian incidence energy normalized incidence energy and randic energy

on the extended estrada index of some graphs abstract let g be a graph on n vertices and 1 2 n the eigenvalues of its extended adjacency matrix the extended estrada index eeex is defined as the sum of the terms e i i 1 2 n in this paper we show the extended estrada index of the complete bipartite graphs and the friendship graphs also the relationship of the extended estrada index between the m splitting m shadow graphs and the original graphs are obtained

bicyclic signed graphs with minimal and second minimal energy abstract let s g be a signed graph of order n and size m where g is its underlying graph and is the sign function a connected signed graph s is said to be bicyclic if m n 1 in this paper we provide characterizations of bicyclic signed graphs with minimal and second minimal energy we notice that there are gaps in the proofs of the main results in j zhang h kan 2014 32 and j zhang b zhou 2005 31 which deal with minimal energy of bicyclic graphs and we plug these gaps as a consequence of the results obtained for bicyclic signed graphs

the weights of the orthogonals of the extended quadratic binary goppa codes starting from results on elliptic curves and kloosterman sums over the finite field ge 2 sup t the authors determine the weights of the orthogonals of some binary linear codes the melas code of length the irreducible cyclic binary code of length 2 sup t 1 and the extended binary goppa codes defined by polynomials of degree two

on the energy of some circulant graphs we give an explicit construction of circulant graphs of very high energy this construction is based on gauss sums we also show the littlewood conjecture can be used to establish new result for a certain class of circulant graphs

the laplacian energy and laplacian estrada index of random multipartite graphs abstract let g be a simple graph on n vertices and m edges and 1 2 n be the eigenvalues of the laplacian matrix of g the laplacian energy of g is defined as e l g i 1 n i 2 m n and the laplacian estrada index of g is defined as l e e g i 1 n e i 2 m n in this paper we establish asymptotic lower and upper bounds to the laplacian energy and laplacian estrada index respectively for random multipartite graphs

the energy of random signed graph abstract a signed graph g is a graph with a sign attached to each of its edges where g is the underlying graph of g the energy of a signed graph g is the sum of the absolute values of the eigenvalues of the adjacency matrix a g of g the random signed graph model g n p q is defined as follows let p q 0 be fixed 0 p q 1 given a set of n vertices between each pair of distinct vertices there is either a positive edge with probability p or a negative edge with probability q or else there is no edge with probability 1 p q the edges between different pairs of vertices are chosen independently in this paper we obtain an exact estimate of energy for almost all signed graphs furthermore we establish lower and upper bounds to the energy of random multipartite signed graphs

the energy of a graph abstract the energy e g of a simple graph g is defined to be the sum of the absolute values of the eigen values of g if g is a k regular graph on n vertices then e g k k n 1 n k b 2 and this bound is sharp it is shown that for each 0 there exist infinitely many n for each of which there exists a k regular graph g of order n with k e g b 2 two graphs with the same number of vertices are equienergetic if they have the same energy we show that for any positive integer n 3 there exist two equienergetic graphs of order 4n that are not cospectral

on the minimal energy of graphs abstract the energy of a graph is the sum of the absolute values of the eigenvalues of the graph which is used to approximate the total electron energy of the molecule in this paper we determine the n e graphs with minimal energy for e n 1 and n 2 which is giving a complete solution to the conjecture for e n 1 and e n 2 proposed by caporossi et al in 4 moreover we determine the graphs with the minimal and second minimal energies for n 1 e 3 n 2 3 and the unique graph with minimal energy for 3 n 5 2 e 2 n 4 among all quasi trees with n vertices and e edges respectively

lower bounds of nikiforov s energy over digraphs abstract the energy of a graph g is defined as e g i 1 n i where 1 2 n are the eigenvalues of the adjacency matrix of g this concept was extended by nikiforov 8 to digraphs as n d i 1 n i where d is a digraph with n vertices and singular values 1 n upper bounds of n were found by kharaghani and tayfeh rezaie 4 in this work we find lower bounds of n over the set of digraphs

topological structural classes of complex networks we use theoretical principles to study how complex networks are topologically organized at large scale using spectral graph theory we predict the existence of four different topological structural classes of networks these classes correspond respectively to highly homogenous networks lacking structural bottlenecks networks organized into highly interconnected modules with low inter community connectivity networks with a highly connected central core surrounded by a sparser periphery and networks displaying a combination of highly connected groups quasicliques and groups of nodes partitioned into disjoint subsets quasibipartites here we show by means of the spectral scaling method that these classes really exist in real world ecological biological informational technological and social networks we show that neither of three network growth mechanisms random with uniform distribution preferential attachment and random with the same degree sequence as real network is able to reproduce the four structural classes of complex networks these models reproduce two of the network classes as a function of the average degree but completely fail in reproducing the other two classes of networks

energy of matrices abstract let m n c denote the space of n n matrices with entries in c we define the energy of a m n c as 1 e a k 1 n k t r a n where 1 n are the eigenvalues of a tr a is the trace of a and z denotes the modulus of z c if a is the adjacency matrix of a graph g then e a is precisely the energy of the graph g introduced by gutman in 1978 in this paper we compare the energy e with other well known energies defined over matrices then we find upper and lower bounds of e which extend well known results for the energies of graphs and digraphs also we obtain new results on energies defined over the adjacency laplacian and signless laplacian matrices of digraphs

asymptotic laplacian energy like invariant of lattices let 1 2 n denote the laplacian eigenvalues of a graph g with n vertices the laplacian energy like invariant denoted by lel g i 1 n 1 i is a novel topological index in this paper we show that the laplacian energy like per vertex of various lattices is independent of the toroidal cylindrical and free boundary conditions simultaneously the explicit asymptotic values of the laplacian energy like in these lattices are obtained moreover our approach implies that in general the laplacian energy like per vertex of other lattices is independent of the boundary conditions

estimating the estrada index abstract let g be a graph on n vertices and let 1 2 n be its eigenvalues the estrada index of g is a recently introduced graph invariant defined as ee i 1 n e i we establish lower and upper bounds for ee in terms of the number of vertices and number of edges also some inequalities between ee and the energy of g are obtained

topology on words we investigate properties of topologies on sets of finite and infinite words over a finite alphabet the guiding example is the topology generated by the prefix relation on the set of finite words considered as a partial order this partial order extends naturally to the set of infinite words hence it generates a topology on the union of the sets of finite and infinite words we consider several partial orders which have similar properties and identify general principles according to which the transition from finite to infinite words is natural we provide a uniform topological framework for the set of finite and infinite words to handle limits in a general fashion

note on the energy of regular graphs abstract for a simple graph g the energy e g is defined as the sum of the absolute values of all the eigenvalues of its adjacency matrix a g let n m respectively be the number of vertices and edges of g one well known inequality is that e g 1 n 1 2 m 1 where 1 is the spectral radius if g is k regular we have e g k k n 1 n k denote e 0 k k n 1 n k balakrishnan r balakrishnan the energy of a graph linear algebra appl 387 2004 287 295 proved that for each 0 there exist infinitely many n for each of which there exists a k regular graph g of order n with k n 1 and e g e 0 and proposed an open problem that given a positive integer n 3 and 0 does there exist a k regular graph g of order n such that e g e 0 1 in this paper we show that for each 0 there exist infinitely many such n that e g e 0 1 moreover we construct another class of simpler graphs which also supports the first assertion that e g e 0

extremal laplacian energy of directed trees unicyclic digraphs and bicyclic digraphs abstract let a g and d g be the adjacency matrix of a digraph g with n vertices and the diagonal matrix of vertex outdegrees of g respectively then the laplacian matrix of the digraph g is l g d g a g the laplacian energy of a digraph g is defined as l e g i 1 n i 2 by using second spectral moment where 1 2 n are all the eigenvalues of l g of g in this paper by using arc shifting operation and out star shifting operation we determine the directed trees unicyclic digraphs and bicyclic digraphs which attain maximal and minimal laplacian energy among all digraphs with n vertices respectively

how many weights can a cyclic code have upper and lower bounds on the largest number of weights in a cyclic code of given length dimension and alphabet are given an application to irreducible cyclic codes is considered sharper upper bounds are given for the special cyclic codes called here strongly cyclic whose nonzero codewords have period equal to the length of the code asymptotics are derived on the function gamma k q that is defined as the largest number of nonzero weights a cyclic code of dimension k over mathbb f q can have and an algorithm to compute it is sketched the nonzero weights in some infinite families of reed muller codes either binary or q ary as well as in the q ary hamming code are determined two difficult results of independent interest

extremality of degree based graph entropies abstract many graph invariants have been used for the construction of entropy based measures to characterize the structure of complex networks based on shannon s entropy we study graph entropies which are based on vertex degrees by using so called information functionals when considering shannon entropy based graph measures there has been very little work to find their extremal values the main contribution of this paper is to prove some extremal values for the underlying graph entropy of certain families of graphs and to find the connection between the graph entropy and the sum of degree powers further conjectures to determine extremal values of graph entropies are given

asymptotic energy of lattices abstract the energy of a simple graph g arising in chemical physics denoted by e g is defined as the sum of the absolute values of eigenvalues of g as the dimer problem and spanning trees problem in statistical physics in this paper we propose the energy per vertex problem for lattice systems in general for a type of lattice in statistical physics to compute the entropy constant with toroidal cylindrical mobius band klein bottle and free boundary conditions are different tasks with different hardness and may have different solutions we show that the energy per vertex of plane lattices is independent of the toroidal cylindrical mobius band klein bottle and free boundary conditions in particular the asymptotic formulae of energies of the triangular 3 3 4 2 and hexagonal lattices with toroidal cylindrical mobius band klein bottle and free boundary conditions are obtained explicitly

energy of unitary cayley graphs and gcd graphs abstract this work is based on ideas of ilic a ilic the energy of unitary cayley graphs linear algebra appl 431 2009 1881 1889 on the energy of unitary cayley graph for a finite commutative ring r with unity 1 0 the unitary cayley graph of r is the cayley graph whose vertex set is r and the edge set is a b a b r and a b r where r is the group of units of r we study the eigenvalues of the unitary cayley graph of a finite commutative ring and some gcd graphs and compute their energy moreover we obtain the energy for the complement of unitary cayley graphs

on the maximal energy tree with two maximum degree vertices abstract for a simple graph g the energy e g is defined as the sum of the absolute values of all eigenvalues of its adjacent matrix for 3 and t 3 denote by t a t or simply t a the tree formed from a path p t on t vertices by attaching 1 p 2 s on each end of the path p t and t b t or simply t b the tree formed from p t 2 by attaching 1 p 2 s on an end of the p t 2 and 2 p 2 s on the vertex next to the end in li et al 2009 16 proved that among trees of order n with two vertices of maximum degree the maximal energy tree is either the graph t a or the graph t b where t n 4 4 3 however they could not determine which one of t a and t b is the maximal energy tree this is because the quasi order method is invalid for comparing their energies in this paper we use a new method to determine the maximal energy tree it turns out that things are more complicated we prove that the maximal energy tree is t b for 7 and any t 3 while the maximal energy tree is t a for 3 and any t 3 moreover for 4 the maximal energy tree is t a for all t 3 but one exception that t 4 for which t b is the maximal energy tree for 5 the maximal energy tree is t b for all t 3 but 44 exceptions that t is both odd and 3 t 89 for which t a is the maximal energy tree for 6 the maximal energy tree is t b for all t 3 but three exceptions that t 3 5 7 for which t a is the maximal energy tree one can see that for most cases of t b is the maximal energy tree 5 is a turning point and 3 and 4 are exceptional cases which means that for all chemical trees whose maximum degrees are at most 4 with two vertices of maximum degree at least 3 t a has maximal energy with only one exception t a 4 4

the weight enumerator of a class of cyclic codes cyclic codes with two zeros and their dual codes have been a subject of study for many years however their weight distributions are known only for a few cases in this paper the weight distributions of the duals of the cyclic codes with two zeros are settled for a few cases the weight distributions of punctured versions of these codes are also determined for several special cases

bicyclic digraphs with maximal energy if d is a digraph with n vertices then the energy of d is defined as e d k 1 n re z k where re z 1 re zn are the real parts of the eigenvalues z 1 z n of d in this paper we solve a problem proposed in khan et al 2015 we find the maximal value of the energy over the set of all bicyclic digraphs b n with n vertices

quadrature rule based bounds for functions of adjacency matrices bounds for entries of matrix functions based on gauss type quadrature rules are applied to adjacency matrices associated with graphs this technique allows to develop inexpensive and accurate upper and lower bounds for certain quantities estrada index subgraph centrality communicability that describe properties of networks

beyond graph energy norms of graphs and matrices abstract in 1978 gutman introduced the energy of a graph as the sum of the absolute values of graph eigenvalues and ever since then graph energy has been intensively studied since graph energy is the trace norm of the adjacency matrix matrix norms provide a natural background for its study thus this paper surveys research on matrix norms that aims to expand and advance the study of graph energy the focus is exclusively on the ky fan and the schatten norms both generalizing and enriching the trace norm as it turns out the study of extremal properties of these norms leads to numerous analytic problems with deep roots in combinatorics the survey brings to the fore the exceptional role of hadamard matrices conference matrices and conference graphs in matrix norms in addition a vast new matrix class is studied a relaxation of symmetric hadamard matrices the survey presents solutions to just a fraction of a larger body of similar problems bonding analysis to combinatorics thus open problems and questions are raised to outline topics for further investigation

network properties revealed through matrix functions the emerging field of network science deals with the tasks of modeling comparing and summarizing large data sets that describe complex interactions because pairwise affinity data can be stored in a two dimensional array graph theory and applied linear algebra provide extremely useful tools here we focus on the general concepts of centrality communicability and betweenness each of which quantifies important features in a network some recent work in the mathematical physics literature has shown that the exponential of a network s adjacency matrix can be used as the basis for defining and computing specific versions of these measures we introduce here a general class of measures based on matrix functions and show that a particular case involving a matrix resolvent arises naturally from graph theoretic arguments we also point out connections between these measures and the quantities typically computed when spectral methods are used for data mining tasks such as clustering and ordering we finish with computational examples showing the new matrix resolvent version applied to real networks

two characterizations of rational adherences resume nous presentons deux caracterisations des adherences en termes de langages finis d homomorphismes strictement alphabetiques et d homomorphismes uniformes inverses nous en deduisons une caracterisation similaire pour les langages rationnels

on the laplacian coefficients of unicyclic graphs abstract let g be a graph of order n and let p g k 0 n 1 k c k n k be the characteristic polynomial of its laplacian matrix generalizing an approach of mohar on graph transformations we show that among all connected unicyclic graphs of order n the k th coefficient c k is largest when the graph is a cycle c n and smallest when the graph is the a s n with an additional edge between two of its pendent vertices a relation to the recently established laplacian like energy of a graph is discussed

remarks on hyperenergetic circulant graphs we first settle an open problem of balakrishnan from linear algebra appl 387 2004 287 295 further if ci n k1 k2 km n n k1 k2 km n 2 ki n for i 1 2 m denotes a circulant graph with the vertex set v 0 1 n 1 such that a vertex u is adjacent to all vertices of v u except u ki mod n i 1 2 m w e show that for any given k1 k2 km almost all circulant graphs ci n k1 k2 km are hyperenergetic

the energy of graphs and matrices abstract given a complex m n matrix a we index its singular values as 1 a 2 a and call the value e a 1 a 2 a the energy of a thereby extending the concept of graph energy introduced by gutman let 2 m n a be an m n nonnegative matrix with maximum entry and a 1 n extending previous results of koolen and moulton for graphs we prove that e a a 1 m n m 1 a 2 2 a 1 2 m n n m m 2 furthermore if a is any nonconstant matrix then e a 1 a a 2 2 1 2 a 2 a finally we note that wigner s semicircle law implies that e g 4 3 o 1 n 3 2 for almost all graphs g

extremal energies of trees with a given domination number abstract the energy of a graph is defined as the sum of the absolute values of the eigenvalues of its adjacency matrix let t n be the set of trees of order n and with domination number in this paper we characterize the tree from t n with the minimal energy and determine the tree from t n where n k with maximal energy for k 2 3 n 4 n 3 n 2

energy of line graphs the energy of a graph is equal to the sum of the absolute values of its eigenvalues the energy of a matrix is equal to the sum of its singular values we establish relations between the energy of the line graph of a graph g and the energies associated with the laplacian and signless laplacian matrices of g

secret sharing schemes from binary linear codes in principle every linear code can be used to construct a secret sharing scheme however in general determining the access structure of the scheme is very hard on the other hand finding error correcting codes that produce secret sharing schemes with efficient access structures is also difficult in this paper we study a set of minimal codewords for certain classes of binary linear codes and then determine the access structure of secret sharing schemes based on these codes furthermore we prove that the secret sharing schemes obtained are democratic in the sense that every participant is involved in the same number of minimal access sets

the skew energy of random oriented graphs abstract given a graph g let g be an oriented graph of g with the orientation and skew adjacency matrix s g the skew energy of the oriented graph g denoted by e s g is defined as the sum of the absolute values of all the eigenvalues of s g in this paper we study the skew energy of random oriented graphs and formulate an exact estimate of the skew energy for almost all oriented graphs by generalizing wigner s semicircle law moreover we consider the skew energy of random regular oriented graphs g n d and get an exact estimate of the skew energy for almost all regular oriented graphs

on the normalized laplacian spectral radius laplacian incidence energy and kemeny s constant abstract let g v e be a simple connected graph of order n and size m and let 1 2 n 1 n 0 be the normalized laplacian eigenvalues of g the greatest eigenvalue 1 is referred to as the normalized laplacian spectral radius the laplacian incidence energy the randic energy and kemeny s constant of g are defined by l i e g i 1 n 1 i r e g i 1 n i 1 and k g i 1 n 1 1 i respectively new lower bounds for the normalized laplacian spectral radius are obtained the acquired bounds are then used to obtain new bounds for the l i e g in addition some relations between l i e g k g r e g are determined

distance powers of unitary cayley graphs let g be a graph and let diam g denote the diameter of g the distance power gn of g is the undirected graph with vertex set v g in which x and y are adjacent if their distance d x y in g belongs to n where n is a non empty subset of 1 2 diam g the unitary cayley graph is the graph having the vertex set z n and the edge set a b a b z n gcd a b n 1 in this paper we determine the energies of distance powers of unitary cayley graphs and classify all ramanujan distance powers of unitary cayley graphs by the energies of distance powers of unitary cayley graphs we construct infinitely many pairs of non cospectral equienergetic graphs moreover we characterize all hyperenergetic distance powers of unitary cayley graphs

applications of recurrence relations for the characteristic polynomials of bethe trees a bethe tree of k levels b k d is a rooted tree such that the root vertex has degree d the vertices from level 2 to k 1 have degree d 1 and the vertices at level k are leaves in this paper we obtain a recurrence relation for the characteristic polynomial and the laplacian characteristic polynomial of bethe trees as an application we prove that there are no integral bethe trees except for the star b 2 n 2 and we obtain a recurrence relation for the laplacian energy like invariant of bethe trees

integral circulant graphs of prime power order with maximal energy the energy of a graph is the sum of the moduli of the eigenvalues of its adjacency matrix we study the energy of integral circulant graphs also called gcd graphs which can be characterized by their vertex count n and a set d of divisors of n in such a way that they have vertex set zn and edge set a b a b zn gcd a b n d using tools from convex optimization we analyze the maximal energy among all integral circulant graphs of prime power order ps and varying divisor sets d our main result states that this maximal energy approximately lies between s p 1 ps 1 and twice this value we construct suitable divisor sets for which the energy lies in this interval we also characterize hyperenergetic integral circulant graphs of prime power order and exhibit an interesting topological property of their divisor sets

supersingular curves over finite fields and weight divisibility of codes motivated by a recent article of the second author we relate a family of artin schreier type curves to a sequence of codes we describe the algebraic structure of these codes and we show that they are quasi cyclic codes we show that if the family of artin schreier type curves consists of supersingular curves then the weights in the related codes are divisible by a certain power of the characteristic we give some applications of the divisibility result including showing that some weights in certain cyclic codes are eliminated in subcodes

on products and line graphs of signed graphs their eigenvalues and energy abstract in this article we examine the adjacency and laplacian matrices and their eigenvalues and energies of the general product non complete extended p sum or neps of signed graphs we express the adjacency matrix of the product in terms of the kronecker matrix product and the eigenvalues and energy of the product in terms of those of the factor graphs for the cartesian product we characterize balance and compute expressions for the laplacian eigenvalues and laplacian energy we give exact results for those signed planar cylindrical and toroidal grids which are cartesian products of signed paths and cycles we also treat the eigenvalues and energy of the line graphs of signed graphs and the laplacian eigenvalues and laplacian energy in the regular case with application to the line graphs of signed grids that are cartesian products and to the line graphs of all positive and all negative complete graphs

degree based energies of graphs abstract let g v e be a simple graph of order n and size m with vertex set v g v 1 v 2 v n without isolated vertices and sequence of vertex degrees d 1 d 2 d n 0 d i d g v i if the vertices v i and v j are adjacent we denote it as v i v j e g or i j with ti we denote a topological index that can be represented as t i t i g i j f d i d j where f is an appropriately chosen function with the property f x y f y x a general extended adjacency matrix a a i j of g is defined as a i j f d i d j if the vertices v i and v j are adjacent and a i j 0 otherwise denote by f i i 1 2 n the eigenvalues of a the energy of the general extended adjacency matrix is defined as e t i e t i g i 1 n f i lower and upper bounds on e t i are obtained by means of the present approach a plethora of earlier established results can be obtained as special cases

the weight distribution of a class of cyclic codes related to hermitian forms graphs the determination of weight distribution of cyclic codes involves the evaluation of gauss sums and exponential sums despite some cases where a neat expression is available the computation is generally rather complicated in this note we determine the weight distribution of a class of reducible cyclic codes whose dual codes may have arbitrarily many zeros this goal is achieved by building an unexpected connection between the corresponding exponential sums and the spectra of hermitian forms graphs

optimal cyclic codes with generalized niho type zeros and the weight distribution in this paper we extend two earlier works further in two directions and compute the weight distribution of these cyclic codes under more relaxed conditions it is interesting to note that many cyclic codes in the family are optimal and have only a few non zero weights besides using similar ideas we carry out some subtle manipulation of certain exponential sums

the triangular kagome lattices revisited abstract the dimer problem ising spins and bond percolation on the triangular kagome lattice have been studied extensively by physicists in this paper based on the fact the triangular kagome lattice with toroidal boundary condition can be regarded as the line graph of 3 12 12 lattice with toroidal boundary condition we derive the formulae of the number of spanning trees the energy and the kirchhoff index of the triangular kagome lattice with toroidal boundary condition

the three subfamilies of rational ω languages closed under ω transduction abstract the transductions defined by gire as bimorphisms coincide with the compositions of non erasing morphisms and inverse non erasing morphisms we prove that there are exactly three subfamilies of rational languages which are closed under transduction the family of rational adherences the family of rational languages whose complementary is deterministic and the whole family of rational languages

3 regular mixed graphs with optimum hermitian energy abstract let g be a simple undirected graph and g be a mixed graph of g with the generalized orientation and hermitian adjacency matrix h g then g is called the underlying graph of g the hermitian energy of the mixed graph g denoted by e h g is defined as the sum of all the singular values of h g a k regular mixed graph on n vertices having hermitian energy n k is called a k regular optimum hermitian energy mixed graph liu and li 2015 11 proposed the problem of determining all the k regular connected optimum hermitian energy mixed graphs this paper is to give a solution to the problem for the case k 3

energy matching number and odd cycles of graphs abstract the energy e g of a graph g is the sum of the absolute values of the eigenvalues of the adjacency matrix of g wong et al 2018 11 proved that e g 2 g where g denotes the matching number of g we settle the equality case that was left open from that paper we also show that for connected graphs without perfect matching e g 2 g 1 except for complete bipartite graphs of the form k r r 1 further for graphs with vertex disjoint cycles we prove that e g 2 g c o g where c o g denotes the number of odd cycles of g with length at least 5 this improves another result of wong et al

on generalized distance energy of graphs abstract for the distance matrix d g and diagonal matrix of the vertex transmissions t r g of a simple connected graph g the generalized distance matrix d g is the convex combinations of t r g and d g and is defined as d g t r g 1 d g for 0 1 if 1 2 n are the eigenvalues of d g we define the generalized distance energy of the graph g as e d g i 1 n i 2 w g n where w g is the wiener index of g this is analogous to the energies associated with the distance laplacian and distance signless laplacian matrices of g we obtain upper and lower bounds for the generalized distance energy of graphs in terms of various parameters associated with the structure of the graph g we show that for 1 2 1 the complete bipartite graph has the minimum generalized distance energy among all connected bipartite graphs and for 0 2 n 3 n 2 the star graph has the minimum generalized distance energy among all trees

the increase in the resolvent energy of a graph due to the addition of a new edge abstract the resolvent energy er g of a graph g on n vertices whose adjacency matrix has eigenvalues 1 n is the sum of the reciprocals of the numbers n 1 n n we introduce the resolvent energy matrix r g and present an algorithm that produces this matrix this algorithm may also be used to update r g when new edges are introduced to g using the resolvent energy matrix r g we determine the increase in the resolvent energy er g of g caused by such edge additions made to g moreover we express this increase in terms of the characteristic polynomial of g and the characteristic polynomials of three vertex deleted subgraphs of g

graphs and matrices with maximal energy abstract given a complex m n matrix a we index its singular values as 1 a 2 a and call the value e a 1 a 2 a the energy of a thereby extending the concept of graph energy introduced by gutman koolen and moulton proved that e g n 2 1 n for any graph g of order n and exhibited an infinite family of graphs with e g v g 2 1 v g we prove that for all sufficiently large n there exists a graph g g n with e g n 3 2 2 n 11 10 this implies a conjecture of koolen and moulton we also characterize all square nonnegative matrices and all graphs with energy close to the maximal one in particular such graphs are quasi random

extremal iota energy of bicyclic digraphs the energy of an n vertex digraph d is defined by e d k 1n re zk where z1 zn are eigenvalues of d and re zk is the real part of eigenvalue zk very recently a new type of energy of digraphs has been introduced which is known as the iota energy of digraphs the iota energy of the digraph d is defined by ec d k 1n im zk where z1 zn are eigenvalues of d and im zk is the imaginary part of eigenvalue zk the unicyclic digraphs with extremal iota energy are known in this paper we consider a class dn of n vertex bicyclic digraphs with vertex disjoint directed cycles and find the digraphs in dn with minimal and maximal iota energy

extremal values of matching energies of one class of graphs in 1978 gutman proposed the concept of graph energy defined as the sum of the absolute values of eigenvalues of the adjacency matrix of a molecular graph which is related to the energy of electrons in conjugated hydrocarbons recently gutman and wagner proposed the concept of matching energy and pointed out that the chemical applications of matching energy go back to the 1970s in this paper we study the extremal values of the matching energy and characterize the graphs with minimal matching energy among all tricyclic graphs with a given diameter our methods can help to find more extremal values for other classes of molecular networks and the results suggest the structures with extremal energies

weight distributions of two classes of cyclic codes with respect to two distinct order elements cyclic codes are an interesting type of linear codes and have wide applications in communication and storage systems due to their efficient encoding and decoding algorithms cyclic codes have been studied for many years but their weight distributions are known only for a few cases in this paper let fr be an extension of a finite field fq and r qm we determine the weight distributions of the cyclic codes c c a b a b fr c a b trr q ag10 bg20 trr q ag1n 1 bg2n 1 g1 g2 fr in the following two cases 1 ord g1 n n r 1 and g2 1 and 2 ord g1 n g2 g12 ord g2 n 2 m 2 and 2 r 1 n q 1

the weight distributions of the duals of cyclic codes with two zeros cyclic codes with two zeros and their dual codes have been a subject of study for many years however their weight distributions are known only for a few cases in this paper the weight distributions of the duals of the cyclic codes with two zeros are settled for a few more cases

the weight distributions of cyclic codes and elliptic curves cyclic codes with two zeros and their dual codes as a practically and theoretically interesting class of linear codes have been studied for many years and find many applications the determination of the weight distributions of such codes is an open problem generally the weight distributions of cyclic codes are difficult to determine utilizing a class of elliptic curves this paper determines the weight distributions of dual codes of q ary cyclic codes with two zeros for a few more cases where q is an odd prime power

upper bounds for some graph energies a general inequality for non negative real numbers is proven based on it upper bounds for ordinary graph energy minimum dominating energy minimum covering energy laplacian energy like invariant laplacian energy randic energy and incidence energy are obtained

a novel method for analyzing inverse problem of topological indices of graphs using competitive agglomeration a novel method for analyzing the inverse problem of topological indices of graph is proposed in this paper which applies the data mining technique to classification of graphs according to topological indices differing from the way of using topological indices to categorize graphs into different isomorphic classes this method aims at dividing molecular graphs with n vertices into several compact classes according to their topological indices then the chemical and physical features of each kind of graph can be researched and new compounds can be found which has great significance on bio medical filed in the experiment three classes of simple connected graphs with 5 6 and 7 vertices respectively are investigated and analyzed using the proposed method the experimental results show the validity of it however there are still some problems needed to be researched further such as selecting the vector of topological indices for clustering selecting the distance between the vectors of topological indices choosing the aspects to analyze the properties of each kind of graph after clustering etc

on the integral weighted oriented unicyclic graphs with minimum skew energy abstract let g be a weighted oriented graph which is obtained from a simple weighted undirected graph by assigning an orientation to each of its edges for a weighted oriented graph g the undirected graph obtained from g by removing the orientation and the weight of each of its arcs is called the underlying graph of g denoted by g by u n m m n we denote the set of all connected integral weighted oriented unicyclic graphs with order n in which each arc is assigned a positive integral weight and the sum of the weights of all arcs is equal to the specified integer m in this paper we investigate the minimal skew energies of integral weighted unicyclic oriented graphs showing that the underlying graph of the oriented graph with minimum skew energy among all graphs over u n m n 6 is s n 3 the graph obtained from a triangle by attaching n 3 pendent edges in exactly one of its vertices moreover we show that its weight sequence has form w 1 a a a k a 1 a 1 a 1 n 3 k w 2 w 3 in which the arc lying on the cycle c 3 and incident to no pendent arcs has weight w 1 and the two largest weights correspond the other two arcs of c 3 where positive integer numbers w 1 w 2 w 3 a and k satisfy 1 w 1 a a 1 w 2 w 3 a w 1 and m w 1 ka n k 3 a 1 w 2 w 3 in addition we determine the weight sequence above for n m 3 n 1

a homomorphic characterization of principal semiafls without using intersection with regular sets abstract if one is asked to consider which one of the semiafl or afl operations differs most from the remaining ones a set theoretically well grounded conclusion would be intersection with regular sets inspired by the result of culik fich and salomaa 1 we shall show that this operation can be replaced by catenation with an endmarker the basis of the paper is the result that any free principal semiafl m l can be presented in the form m l h 4 h 3 1 h 2 h 1 1 l each h 1 a homomorphism h 2 and h 3 free h 4 length preserving where is a symbol not in the alphabet of l if m l is not free we get the same representation but now h 4 erases the last letter from each word in h 3 1 h 2 1 h 1 1 l and is length preserving for other letters for the family of free regular languages we may choose l the last section contains a homomorphic characterization of regular star languages

optimum skew energy of a tournament abstract let g be a simple undirected graph and g the corresponding oriented graph of g with the orientation the skew energy of g denoted by e g is defined as the sum of the singular values of its skew adjacency matrix s g in 2010 adiga et al proved e g n where is the maximum degree of g of order n in this paper we characterize the skew energy of a tournament and present some properties about an optimum skew energy tournament

on the tree with diameter 4 and maximal energy the energy of a simple graph is defined as the sum of the absolute values of all the eigenvalues of its adjacency matrix jianping found the structure of trees which have maximal energy among all trees of diameter 4 whose centers are of degree t this paper generalizes the result we determine the structure of the general tree of diameter 4 and maximal energy

on the energy of 0 1 matrices the energy of a matrix is the sum of its singular values we study the energy of 0 1 matrices and present two methods for constructing balanced incomplete block designs whose incidence matrices have the maximum possible energy amongst the family of all 0 1 matrices of given order and total number of ones we also find a new upper bound for the energy of p q bipartite graphs

the influence of miroslav fiedler on spectral graph theory abstract this note is a write up of a talk given at the ilas meeting in braunschweig 2011 at the minisymposium celebrating the 80th birthday of miroslav fiedler the purpose of the talk is to outline the impact of fiedler s work on the development of spectral graph theory fiedler is best known for putting forward the algebraic connectivity and its eigenvector these two topics were genuine gold strikes which have motivated thousands of studies in pure and applied science but there is more in fiedler s work that influences spectral graph theory and still more waiting to be discovered

x automata on ω words abstract for any storage type x the languages accepted by x automata are investigated six accepting conditions including those introduced by landweber are compared for x automata the inclusions between the corresponding six families of languages are essentially the same as for finite state automata apart from unrestricted automata also real time and deterministic automata are considered the main tools for this investigation are 1 a characterization of the languages accepted by x automata in terms of inverse x transductions of finite state languages and 2 the existence of topological upper bounds on some of the families of accepted languages independent of the storage type x

quadratic unitary cayley graphs of finite commutative rings abstract the purpose of this paper is to study spectral properties of a family of cayley graphs on finite commutative rings let r be such a ring and r be its set of units let q r u 2 u r and t r q r q r we define the quadratic unitary cayley graph of r denoted by g r to be the cayley graph on the additive group of r with respect to t r that is g r has vertex set r such that x y r are adjacent if and only if x y t r it is well known that any finite commutative ring r can be decomposed as r r 1 r 2 r s where each r i is a local ring with maximal ideal m i let r 0 be a local ring with maximal ideal m 0 such that r 0 m 0 3 mod 4 we determine the spectra of g r and g r 0 r under the condition that r i m i 1 mod 4 for 1 i s we compute the energies and spectral moments of such quadratic unitary cayley graphs and determine when such a graph is hyperenergetic or ramanujan

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

remarks on the energy of regular graphs abstract the energy of a graph is the sum of the absolute values of the eigenvalues of its adjacency matrix this note is about the energy of regular graphs it is shown that graphs that are close to regular can be made regular with a negligible change of the energy also a k regular graph can be extended to a k regular graph of a slightly larger order with almost the same energy as an application it is shown that for every sufficiently large n there exists a regular graph g of order n whose energy g satisfies g 1 2 n 3 2 n 13 10 several infinite families of graphs with maximal or submaximal energy are given and the energy of almost all regular graphs is determined

some properties of graphs constructed from 2 designs abstract let d p b be any nontrivial nonsymmetric 2 v k design in this paper we construct a connected regular and bipartite graph say ig v b d from d some graphic properties of ig v b d are investigated in particular the results in the paper bridges the equivalency between edge transitivity of ig v b d and flag transitivity of d with the equivalency it is obtained that ig v b d is a semisymmetric graph as long as d is a nonsymmetric flag transitive 2 v k design

on the weight distributions of two classes of cyclic codes let q pm where p is an odd prime mges2 and 1lesklesm 1 let tr be the trace mapping from fq to fp and zetap e2pii p be a primitive pth root of unity in this paper we determine the value distribution of the following exponential sums sigmaxisinf qchi alphaxp k 1 betax2 alpha betaisinfq where chi x zetap tr x is the canonical additive character of fq as applications we have the following 1 we determine the weight distribution of the cyclic codes c1 and c2 over fpt with parity check polynomial h2 x h3 x and h1 x h2 x h3 x respectively where t is a divisor of d gcd m k and h1 x h2 x and h3 x are the minimal polynomials of pi 1 pi 2 and pi p k 1 over fpt respectively for a primitive element pi of fq 2 we determine the correlation distribution between two m sequences of period q 1 moreover we find a new class of p ary bent functions this paper extends the results in feng and luo 2008