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

hermitian adjacency spectrum and switching equivalence of mixed graphs abstract it is shown that an undirected graph g is cospectral with the hermitian adjacency matrix of a mixed graph d obtained from a subgraph h of g by orienting some of its edges if and only if h g and d is obtained from g by a four way switching operation if g is connected this happens if and only if 1 g 1 d all mixed graphs of rank 2 are determined and this is used to classify which mixed graphs of rank 2 are cospectral with respect to their hermitian adjacency matrix several families of mixed graphs are found that are determined by their hermitian spectrum in the sense that they are cospectral precisely to those mixed graphs that are switching equivalent to them

spectra of digraphs in this primarily expository paper we survey classical and some more recent results on the spectra of digraphs equivalently the spectra of 0 1 matrices with emphasis on the spectral radius

bounds for the matching number and cyclomatic number of a signed graph in terms of rank 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 let m g c g and r g be the matching number the cyclomatic number and the rank of the adjacency matrix of g respectively in this paper we investigate the relation among the rank the matching number and the cyclomatic number of a signed graph and prove that 2 m g 2 c g r g 2 m g c g furthermore signed graphs reaching the lower bound or the upper bound are respectively characterized

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

hermitian laplacian matrix and positive of mixed graphs a mixed graph is obtained from an undirected graph by orienting a subset of its edges the hermitian adjacency matrix of a mixed graph m of order n is the n n matrix h m h k l where h k l h l k i i 1 if there exists an orientation from vk to vl and h k l h l k 1 if there exists an edge between vk and vl but not exist any orientation and h k l 0 otherwise the value of a mixed walk w v 1 v 2 v 3 v l is h w h 12 h 23 h l 1 l a mixed walk is positive negative if h w 1 h w 1 a mixed cycle is called positive if its value is 1 a mixed graph is positive if each of its mixed cycle is positive in this work we firstly present the necessary and sufficient conditions for the positive of a mixed graph secondly we introduce the incident matrix and hermitian laplacian matrix of a mixed graph and derive some results about the hermitian laplacian spectrum

interlacing families and the hermitian spectral norm of digraphs abstract it is proved that for any finite connected graph g there exists an orientation of g such that the spectral radius of the corresponding hermitian adjacency matrix is smaller or equal to the spectral radius of the universal cover of g with equality if and only if g is a tree this provides a suitable answer to a problem proposed by mohar the proof uses the method of interlacing families of polynomials that was developed by marcus spielman and srivastava in their seminal work on the existence of infinite families of ramanujan graphs

mixed paths and cycles determined by their spectrum abstract a mixed graph is obtained from a graph by orienting some of its edges the hermitian adjacency matrix of a mixed graph with the vertex set v 1 v n is the matrix h h i j n n where h i j h j i i if there is a directed edge from v i to v j h i j 1 if there exists an undirected edge between v i and v j and h i j 0 otherwise the hermitian spectrum of a mixed graph is defined to be the spectrum of its hermitian adjacency matrix in this paper we study mixed graphs which are determined by their hermitian spectrum dhs first we show that each mixed cycle is switching equivalent to either a mixed cycle with no directed edges c n a mixed cycle with exactly one directed edge c n 1 or a mixed cycle with exactly two consecutive directed edges with the same direction c n 2 and we determine the spectrum of these three types of cycles next we characterize all dhs mixed paths and mixed cycles we show that all mixed paths of even order except p 8 and p 14 are dhs it is also shown that mixed paths of odd order except p 3 are not dhs also all cospectral mates of p 8 p 14 and p 4 k 1 and two families of cospectral mates of p 4 k 3 where k 1 are introduced finally we show that the mixed cycles c 2 k and c 2 k 2 where k 3 are not dhs but the mixed cycles c 4 c 4 2 c 2 k 1 c 2 k 1 2 c 2 k 1 1 and c 2 j 1 except c 7 1 c 9 1 c 12 1 and c 15 1 are dhs where k 1 and j 2

a short proof of tutte s characterization of totally unimodular matrices we give in terms of totally unimodular matrices a short and easy proof of tutte s characterization of regular matroids

the skew energy of a digraph we are interested in the energy of the skew adjacency matrix of a directed graph d which is simply called the skew energy of d in this paper properties of the skew energy of d are studied in particular a sharp upper bound for the skew energy of d is derived in terms of the order of d and the maximum degree of its underlying undirected graph an infinite family of digraphs attaining the maximum skew energy is constructed moreover the skew energy of a directed tree is independent of its orientation and interestingly it is equal to the energy of the underlying undirected tree skew energies of directed cycles under different orientations are also computed some open problems are presented

the spectral radius of trees on k pendant vertices abstract in this paper we consider the following problem of all trees of order n with k pendant vertices n k fixed which achieves the maximal spectral radius we show that the maximal spectral radius is obtained uniquely at t n k where t n k is a tree obtained from a star k 1 k and k paths of almost equal lengths by joining each pendant vertex of k 1 k to an end vertex of one path we also discuss the spectral radius of t n k and get some results

differential equations in the spectral parameter we determine all the potentialsv x for the schrodinger equation 2 v x k2 such that some family of eigenfunctions satisfies a differential equation in the spectral parameterk of the formb k k o x o for each suchv x we determine the algebra of all possible operatorsb and the corresponding functions x

the spectral distribution of random mixed graphs abstract let g be a mixed graph with n vertices h g the hermitian adjacency matrix of g and 1 g 2 g n g the eigenvalues of h g the hermitian energy of g is defined as e h g i 1 n i g in this paper we characterize the limiting spectral distribution of the hermitian adjacency matrices of random mixed graphs and as an application we give an estimation of the hermitian energy for almost all mixed graphs

characterizing the mixed graphs with exactly one positive eigenvalue and its application to mixed graphs determined by their h spectra abstract a mixed graph is obtained by orienting some edges of a simple graph the positive inertia index of a mixed graph is defined as the number of positive eigenvalues of its hermitian adjacency matrix including multiplicities in this paper we study the positive inertia indices of mixed graphs our main results provide a characterization of mixed graphs with positive inertia index 1 as an application some classes of mixed graphs determined by their h spectra are fixed

spacing ratio characterization of the spectra of directed random networks previous literature on random matrix and network science has traditionally employed measures derived from nearest neighbor level spacing distributions to characterize the eigenvalue statistics of random matrices this approach however depends crucially on eigenvalue unfolding procedures which in many situations represent a major hindrance due to constraints in the calculation especially in the case of complex spectra here we study the spectra of directed networks using the recently introduced ratios between nearest and next to nearest eigenvalue spacing thus circumventing the shortcomings imposed by spectral unfolding specifically we characterize the eigenvalue statistics of directed erd ifmmode mbox h o else h o fi s r enyi er random networks by means of two adjacency matrix representations namely 1 weighted non hermitian random matrices and 2 a transformation on non hermitian adjacency matrices which produces weighted hermitian matrices for both representations we find that the distribution of spacing ratios becomes universal for a fixed average degree in accordance with undirected random networks furthermore by calculating the average spacing ratio as a function of the average degree we show that the spectral statistics of directed er random networks undergoes a transition from poisson to ginibre statistics for model 1 and from poisson to gaussian unitary ensemble statistics for model 2 eigenvector delocalization effects of directed networks are also discussed

eigenvalues and colorings of digraphs abstract wilf s eigenvalue upper bound on the chromatic number is extended to the setting of digraphs the proof uses a generalization of brooks theorem to digraph colorings

sturmian words structure combinatorics and their arithmetics abstract we prove some new results concerning the structure the combinatorics and the arithmetics of the set per of all the words w having two periods p and q p q which are coprimes and such that w p q 2 a basic theorem relating per with the set of finite standard sturmian words was proved in de luca and mignosi 1994 the main result of this paper is the following simple inductive definition of per the empty word belongs to per if w is an already constructed word of per then also aw and bw belong to per where denotes the operator of palindrome left closure i e it associates to each word u the smallest palindrome word u having u as a suffix we show that by this result one can construct in a simple way all finite and infinite standard sturmian words we prove also that up to the automorphism which interchanges the letter a with the letter b any element of per can be codified by the irreducible fraction p q this allows us to construct for any n 0 a natural bijection that we call farey correspondence of the set of the farey series of order n 1 and the set of special elements of length n of the set st of all finite sturmian words finally we introduce the concepts of farey tree and farey monoid this latter is obtained by defining a suitable product operation on the developments in continued fractions of the set of all irreducible fractions p q

noncommutative bispectral darboux transformations we prove a general theorem establishing the bispectrality of noncommutative darboux transformations it has a wide range of applications that establish bispectrality of such transformations for differential difference and q difference operators with values in all noncommutative algebras all known bispectral darboux transformations are special cases of the theorem using the methods of quasideterminants and the spectral theory of matrix polynomials we explicitly classify the set of bispectral darboux transformations from rank one differential operators and airy operators with values in matrix algebras these sets generalize the classical calogero moser spaces and wilson s adelic grassmannian

rational solutions of nonlinear evolution equations vertex operators and bispectrality abstract we prove that if q x t 2 t m and r x t 2 t m are certain rational solutions of the akns hierarchy then there are eigenfunctions x k of the akns zs operator l x q r x satisfying a differential equation in the spectral parameter k of the form b k k x where b k k is a matrix differential operator independent of x and is a nonconstant function of x we also discuss the relation between this result and a similar one for the rational solutions of the schrodinger operator with potentials in the manifold of rational solutions of the kdv hierarchy

relations between the inertia indices of a mixed graph and those of its underlying graph abstract given a graph g the mixed graph d g is obtained from g by orienting some of its edges where g is called the underlying graph of d g let p d g n d g resp p g n g be the positive inertia index and negative inertia index of d g resp g in this paper we first establish the inequalities d g p d g p g d g and d g n d g n g d g where d g is the dimension of cycle space of g furthermore all the corresponding extremal graphs are characterized

on the mixed adjacency matrix of a mixed graph a mixed graph is a graph with edges and arcs which can be considered as a hybrid of an undirected graph and a directed graph in this paper we define the mixed adjacency matrix and the mixed energy of a mixed graph the mixed adjacency matrix generalizes both the adjacency matrix of an undirected graph and the skew adjacency matrix of a digraph then we compute the characteristic polynomial of the mixed adjacency matrix of a mixed graph and deduce some basic results from it furthermore we give bounds to the mixed energy of a general mixed graph and we compute the mixed energy of some special mixed graphs at the end of the paper we introduce mixed unitary cayley graphs and compute their spectra

on the characteristic polynomials and h ranks of the weighted mixed graphs abstract let d g be an n vertex weighted mixed graph with hermitian adjacency matrix h d g the characteristic polynomial of the weighted mixed graph d g is defined as d g det i n h d g r 0 n r n r and the h rank of d g written as r k d g is the rank of h d g in this paper we begin by interpreting all the coefficients of the characteristic polynomial d g then we establish recurrences for the characteristic polynomial d g by these obtained results as a unified approach some main results obtained in electron j combin 18 1 2011 156 linear algebra appl 436 2012 3597 3607 linear algebra appl 466 2015 182 207 can be deduced consequently furthermore for a weighted mixed bipartite cactus graph d g we give a necessary and sufficient condition for r k d g 2 t where t is any integer no greater than the matching number of g finally as an application of our results we study the minimum h rank problem determining the minimum h rank of bipartite cactus graphs

hermitian normalized laplacian matrix for directed networks abstract in this paper we extend and generalize the spectral theory of undirected networks towards directed networks by introducing the hermitian normalized laplacian matrix for directed networks in order to start we discuss the courant fischer theorem for the eigenvalues of hermitian normalized laplacian matrix based on the courant fischer theorem we obtain a similar result towards the normalized laplacian matrix of undirected networks for each i 1 2 n any eigenvalue of hermitian normalized laplacian matrix i 0 2 moreover we prove some special conditions if 0 or 2 is an eigenvalue of the hermitian normalized laplacian matrix l x on top of that we investigate the symmetry of the eigenvalues of l x and the edge version for the eigenvalue interlacing result finally we present two expressions for the coefficients of the characteristic polynomial of the hermitian normalized laplacian matrix as an outlook we sketch some novel and intriguing problems to which our apparatus could generally be applied

on a zero curvature problem related to the zs akns operator all the zs akns operators l i x g x r x x which admit two linearly independent families of eigenfunctions satisfying a differential equation in the spectral parameter of the form a v b v x v are characterized here a c b and take values in the space of 2 2 complex matrices

subspaces with well scaled frames abstract in this paper classes of subspaces are introduced that generalize the regular unimodular subspaces of rational coordinate space these spaces are shown to be of interest in the study of linear and integer programming fundamental properties of standard representative matrices for classes of subspaces are established certain subspaces are shown to be projectively equivalent to regular subspaces well known examples of totally unimodular matrices arising from graphs and intervals of the real line are generalized in this context a generalization of heller s theorem bounding the number of distinct columns of a rank r totally unimodular matrix is established in this setting

episturmian words and some constructions of de luca and rauzy abstract in this paper we study infinite episturmian words which are a natural generalization of sturmian words to an arbitrary alphabet a characteristic property is they are closed under reversal and have at most one right special factor of each length they are first obtained by a construction due to de luca which utilizes the palindrome closure they can also be obtained by the way of extended rauzy rules

aperiodic pseudorandom number generators based on infinite words in this paper we study how certain families of aperiodic infinite words can be used to produce aperiodic pseudorandom number generators prngs with good statistical behavior we introduce the well distributed occurrences welldoc combinatorial property for infinite words which guarantees absence of the lattice structure defect in related pseudorandom number generators an infinite word u on a d ary alphabet has the welldoc property if for each factor w of u positive integer m and vector v z m d there is an occurrence of w such that the parikh vector of the prefix of u preceding such occurrence is congruent to v modulo m the parikh vector of a finite word v over an alphabet a has its i th component equal to the number of occurrences of the i th letter of a in v we prove that sturmian words and more generally arnoux rauzy words and some morphic images of them have the welldoc property using the testu01 11 and practrand 5 statistical tests we moreover show that not only the lattice structure is absent but also other important properties of prngs are improved when linear congruential generators are combined using infinite words having the welldoc property

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

complete monotonicity for inverse powers of some combinatorially defined polynomials we prove the complete monotonicity on 0 infty n for suitable inverse powers of the spanning tree polynomials of graphs and more generally of the basis generating polynomials of certain classes of matroids this generalizes a result of szeg and answers among other things a long standing question of lewy and askey concerning the positivity of taylor coefficients for certain rational functions our proofs are based on two ab initio methods for proving that p beta is completely monotone on a convex cone c the determinantal method and the quadratic form method these methods are closely connected with harmonic analysis on euclidean jordan algebras or equivalently on symmetric cones we furthermore have a variety of constructions that given such polynomials can create other ones with the same property among these are algebraic analogues of the matroid operations of deletion contraction direct sum parallel connection series connection and 2 sum the complete monotonicity of p beta for some beta 0 can be viewed as a strong quantitative version of the half plane property hurwitz stability for p and is also related to the rayleigh property for matroids

matroids over partial hyperstructures abstract we present an algebraic framework which simultaneously generalizes the notion of linear subspaces matroids valuated matroids oriented matroids and regular matroids to do this we first introduce algebraic objects which we call tracts they generalize both hyperfields in the sense of krasner and partial fields in the sense of semple and whittle we then define matroids over tracts in fact there are at least two natural notions of matroid in this general context which we call weak and strong matroids we give cryptomorphic axiom systems for such matroids in terms of circuits grassmann plucker functions and dual pairs and establish some basic duality results we then explore sufficient criteria for the notions of weak and strong matroids to coincide this is the case whenever vectors and covectors are orthogonal and is closely related to the notion of perfect fuzzy rings from 15 for example if f is a particularly nice kind of tract called a doubly distributive partial hyperfield we show that the notions of weak and strong matroid over f coincide our theory of matroids over tracts is closely related to but more general than matroids over fuzzy rings in the sense of dress and dress wenzel 12 13 14 15

3 regular digraphs with optimum skew energy abstract the skew energy of a digraph d is defined as the sum of the singular values of its skew adjacency matrix s d in this paper we first interpret the entries of the power of the skew adjacency matrix of a digraph in terms of the number of its walks and then focus on the question posed by adiga et al c adiga r balakrishnan wasin so the skew energy of a graph linear algebra appl 432 2010 1825 1835 of determining all 3 regular connected digraphs with optimum skew energy

singularity of hermitian quasi laplacian matrix of mixed graphs a mixed graph is obtained from an undirected graph by orienting a subset of its edges the hermitian adjacency matrix of a mixed graph m of order n is an n n matrix h m h k l where h k l h l k i i 1 if there exists an orientation from vk to vl and h k l h l k 1 if there exists an edge between vk and vl but not exist any orientation and h k l 0 otherwise let d m d i a g d 1 d 2 d n be a diagonal matrix where di is the degree of vertex vi in the underlying graph mu hermitian matrices l m d m h m q m d m h m are said as the hermitian laplacian matrix hermitian quasi laplacian matrix of mixed graph m respectively in this paper it is shown that they are positive semi definite moreover we characterize the singularity of them in addition an expression of the determinant of the hermitian quasi laplacian matrix is obtained

a bound on the spectral radius of graphs abstract let g be a simple connected graph with n vertices and m edges let a be the adjacency matrix of g the spectral radius a of a satisfies a 2m n 1 with equality if and only if g is isomorphic to one of the following two graphs a the star k1 n 1 b the complete graph kn

a new kind of hermitian matrices for digraphs abstract in an earlier work the author together with guo 7 introduced the hermitian adjacency matrix of directed and partially directed graphs however it appears that a more natural hermitian matrix exists and it is the purpose of this note to bring this new hermitian matrix to the attention of researchers in algebraic graph theory

4 regular oriented graphs with optimum skew energy abstract let g be a simple undirected graph and 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 characterize the underlying graphs of all 4 regular oriented graphs with optimum skew energy and give orientations of these underlying graphs such that the skew energy of the resultant oriented graphs indeed attain optimum it should be pointed out that there are infinitely many 4 regular connected optimum skew energy oriented graphs while the 3 regular case only has two graphs k 4 the complete graph on 4 vertices and q 3 the hypercube

hermitian adjacency matrices and hermitian energies of mixed graphs abstract a complex adjacency matrix of a mixed graph is introduced in the present paper which is a hermitian matrix and called the hermitian adjacency matrix it incorporates both adjacency matrix of an undirected graph and skew adjacency matrix of an oriented graph some of its properties are studied furthermore properties of its characteristic polynomial are studied cospectral problems among mixed graphs including mixed graphs and their underlying graphs oriented graphs and their underlying graphs are studied we give equivalent conditions for a mixed graph especially oriented graph that share the same spectrum with its underlying graph as a consequence we reconfirm a conjecture which was proposed by cui and hou in ref 8 we also show that the spectrum of the hermitian matrix of a mixed graph is invariant when changing the value of any its cut edge if any correspondingly an energy of a mixed graph is introduced and called the hermitian energy it incorporates both the energy of an undirected graph and the skew energy of an oriented graph some of its bounds are given especially the mixed graphs with optimal upper bound of hermitian energy are characterized an infinite family of mixed graphs attaining the maximum hermitian energy is constructed moreover the hermitian energy of a mixed tree is showed to be equal to the energy of its underlying tree finally the integral formula for hermitian energy of a mixed graph is given

latroids and their representation by codes over modules it has been known for some time that there is a connection between linear codes over fields and matroids represented over fields in fact a generator matrix for a linear code over a field is also a representation of a matroid over that field there are intimately related operations of deletion contraction minors and duality on both the code and the matroid the weight enumerator of the code is an evaluation of the tutte polynomial of the matroid and a standard identity relating the tutte polynomials of dual matroids gives rise to a macwilliams identity relating the weight enumerators of dual codes more recently codes over rings and modules have been considered and macwilliams type identities have been found in certain cases in this paper we consider codes over rings and modules with code duality based on a morita duality of categories of modules to these we associate latroids defined here we generalize notions of deletion contraction minors and duality on both codes and latroids and examine all natural relations among these we define generating functions associated with codes and latroids and prove identities relating them generalizing above mentioned generating functions and identities

principal minor version of matrix tree theorem for mixed graphs in yu etal 2017 an analytical expression of the determinant of the hermitian quasi laplacian matrix of mixed graphs has been proven in this paper we are going to extend those results and derive an analytical expression for the principal minors of the hermitian quasi laplacian matrix which is the principal minor version of the matrix tree theorem

differential equations in the spectral parameter for matrix differential operators we use matrix darboux transformation to generate a class of matrix differential operators l that have the following property there exist a family x k of eigenfunctions of l also satisfying a differential equation in the spectral parameter k of the form b k k x whereb k k is a diffrential and is a non constant function of x we also obtain nth order scalar operators which have this property

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 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

monodromy preserving deformation of linear ordinary differential equations with rational coefficients iii abstract a unified treatment of monodromy and spectrum preserving deformations is presented in particular a general procedure is described to reduce the latter into the former consistently the concept of the function previously introduced for the former 2 is extended to the isospectral context it is shown that the general monodromy and spectrum preserving deformation equations can be written as hirota s bilinear differential equations by using the functions as dependent variables

mixed graphs with h rank 3 abstract in the article mohar 2016 9 mohar determined all mixed graphs with h rank 2 and used it to classify cospectral graphs with respect to their hermitian adjacency matrix constructing a class of graphs which can not be determined by their h spectrum in the present paper we investigate the h rank of mixed graphs further determining the h ranks of those mixed graphs with trees cycles and complete bipartite graphs as underlying graphs respectively moreover we characterize all mixed graphs with h rank 3 and show that all connected mixed graphs with h rank 3 can be determined by their h spectrum

matrix valued bispectral operators and quasideterminants we consider a matrix valued version of the bispectral problem that is find differential operators and with matrix coefficients such that there exists a family of matrix valued common eigenfunctions x z where f and are matrix valued functions using quasideterminants we prove that the operators l obtained by non degenerated rational matrix darboux transformations from are bispectral operators where and d is a diagonal matrix we also give a procedure to find an explicit formula for the operator b extending previous results in the scalar case

spectral radius of digraphs with given dichromatic number abstract let d be a digraph with vertex set v d a partition of v d into k acyclic sets is called a k coloring of d the minimum integer k for which there exists a k coloring of d is the dichromatic number d of the digraph d denote g n k the set of the digraphs of order n with the dichromatic number k 2 in this note we characterize the digraph which has the maximal spectral radius in g n k our result generalizes the result of 8 by feng et al

on the skew energy of orientations of hypercubes abstract let d be a digraph of order n and 1 2 n denote all the eigenvalues of the skew adjacency matrix of d the skew energy e s d of d is defined as e s d i 1 n i in this paper it is proved that for any positive integer k 3 there exists a k regular graph of order n having an orientation d with e s d n k this work positively answers a problem proposed by adiga et al c adiga r balakrishnan wasin so the skew energy of a digraph linear algebra appl 432 2010 1825 1835 in addition a digraph is also constructed such that its skew energy is the same as the energy of its underlying graph

proving the deterministic period breaking of linear congruential generators using two tile quasicrystals we describe the design of a family of aperiodic prngs aprngs we show how a one dimensional two tile cut and project quasicrystal 2tqc used in conjunction with lcgs in an aprng generates an infinite aperiodic pseudorandom sequence in the suggested design any 2tqc corresponding to unitary quadratic pisot number combined with either one or two different lcgs can be used

differential equations in the spectral parameter darboux transformations and a hierarchy of master symmetries for kdv we study a certain family of schrodinger operators whose eigenfunctions satisfy a differential equation in the spectral parameter of the formb x we show that the flows of a hierarchy of master symmetries for kdv are tangent to the manifolds that compose the strata of this class ofbispectral potentials this extends and complements a result of duistermaat and grunbaum concerning a similar property for the adler and moser potentials and the flows of the kdv hierarchy

on unimodular graphs abstract we study graphs whose adjacency matrices have determinant equal to 1 or 1 and characterize certain subclasses of these graphs graphs whose adjacency matrices are totally unimodular are also characterized for bipartite graphs having a unique perfect matching we provide a formula for the inverse of the corresponding adjacency matrix and address the problem of when that inverse is diagonally similar to a nonnegative matrix special attention is paid to the case that such a graph is unicyclic

the rank of a complex unit gain graph in terms of the matching number abstract a complex unit gain graph or t gain graph is a triple g t or g for short consisting of a simple graph g as the underlying graph of g the set of unit complex numbers t z c z 1 and a gain function e t with the property that e i j e j i 1 in this paper we prove that 2 m g 2 c g r g 2 m g c g where r g m g and c g are the rank of the hermitian adjacency matrix h g the matching number and the cyclomatic number of g respectively furthermore the complex unit gain graphs g t with r g 2 m g 2 c g and r g 2 m g c g are characterized these results generalize the corresponding known results about undirected graphs mixed graphs and signed graphs moreover we show that 2 m g v 0 r g 2 m g s holds for any s v g such that g s is bipartite and any subset v 0 of v g such that g v 0 is acyclic