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

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

complex unit gain bicyclic graphs with rank 2 3 or 4 abstract a t gain graph is a triple g t consisting of a graph g v e the circle group t z c z 1 and a gain function e t such that e i j e j i 1 e j i the rank of t gain graph denoted by r is the rank of the adjacency matrix of yu et al 2015 8 obtained some properties of inertia of a t gain graph they characterized the t gain unicyclic graphs with small positive or negative index motivated by above in this paper we characterize the complex unit gain connected bicyclic graphs with rank 2 3 or 4

extremal values of energy over oriented bicyclic graphs abstract graph energy can be extended to digraphs via the trace norm the trace norm of a digraph is the trace norm of its adjacency matrix i e the sum of its singular values in this work we find the oriented graphs that attain minimal and maximal trace norm over the set of oriented bicyclic graphs

on the non existence of undesirable equilibria of godard blind equalizers existing results in the literature have proved that particular blind equalization algorithms including godard algorithms are globally convergent in an ideal and nonimplementable setting where a doubly infinite dimensional equalizer is available for adaptation contrary to popular conjectures it is shown that implementable finite dimensional equalizers which attempt to approximate the ideal setting generally fail to have global convergence to acceptable equalizer parameter settings without the use of special remedial measures a theory based on the channel convolution matrix nullspace is proposed to explain the failure of godard algorithms for such practical blind equalization situations this nullspace theory is supported by a simple example showing ill convergence of the godard algorithm

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

direct blind multiuser detection for cdma in multipath without channel estimation we consider the blind multiuser detection problem for asynchronous ds cdma systems operating in a multipath environment using only the spreading code of the desired user we first estimate the column vector subspace of the channel matrix by multiple linear prediction then zero forcing detectors and mmse detectors with arbitrary delay can be obtained without explicit channel estimation this avoids any channel estimation error and the resulting methods are therefore more robust and more accurate corresponding batch algorithms and adaptive algorithms are developed the new algorithms are extremely near far resistant simulations demonstrate the effectiveness of these methods

skew spectra of graphs without even cycles abstract the study of the invariance of the skew spectrum of an oriented graph under different orientations leads to the special family g of graphs without even cycles basic graphical and spectral properties of g are observed as a result the hosoya index of a graph in g can be computed via its skew spectrum we also focus on friendship graphs a subfamily of g in particular the condition under which the skew spectrum of a friendship graph is actually i times the spectrum of some graph is determined where i 1

the number of the skew eigenvalues of digraphs and their relationship with optimum skew energy abstract let g be an oriented graph of simple graph g with the orientation and vertex set v g v 1 v 2 v n the skew adjacency matrix of g is the 0 1 1 matrix s s g s i j such that s i j 1 if v i v j is an arc in g s i j 1 if v j v i is an arc in g and s i j 0 otherwise in this paper all oriented graphs with two and three skew eigenvalues are characterized also we determine the relationship between the number of skew eigenvalues and optimum skew energy of oriented graphs

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

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

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

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

linear prediction methods for blind fractionally spaced equalization we describe adaptive methods for estimating fir zero forcing blind equalizers with arbitrary delay directly from the linear predictions of the observations while most current methods require inversion or singular value decomposition svd of the correlation matrix our methods need only to solve two linear prediction problems they can be implemented as rls or lms algorithms to recursively update the equalizer estimation they are computationally efficient the computational complexity in each recursion can be less than 15 ln sup 2 in the rls case where ln equals the equalizer length and 3l ln in the lms case where l is the number of subchannels the performance of the proposed methods and comparisons with existing approaches are shown by simulation to demonstrate their effectiveness

qr factorization based blind channel identification with second order statistics most eigenstructure based blind channel identification and equalization algorithms with second order statistics need svd or evd of the correlation matrix of the received signal in this paper we address new algorithms based on qr factorization of the received signal directly without calculating the correlation matrix this renders the qr factorization based algorithms more robust against ill conditioned channels i e those channels with almost common zeros among the subchannels first we present a block algorithm that performs the qr factorization of the received data matrix as a whole then a recursive algorithm is developed based on the qr factorization by updating a rank revealing ulv decomposition compared with existing algorithms in the same category our algorithms are computationally more efficient the computation in each recursion of the recursive algorithm is on the order of o m sup 2 if only equalization is required where m is the dimension of the received signal vector our recursive algorithm preserves the fast convergence property of the subspace algorithms thus converging faster than other adaptive algorithms such as the super exponential algorithm with comparable computational complexities moreover our proposed algorithms do not require noise variance estimation numerical simulations demonstrate the good performance of the proposed algorithms

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

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

upper bound of skew energy of an oriented graph in terms of its skew rank abstract let g be an oriented graph with skew adjacency matrix s g the skew energy e s g of g is the sum of the norms of all eigenvalues of s g and the skew rank r s g of g is the rank of s g recently tian and wong gave a lower bound of e s g in terms of its skew rank they proved that r s g e s g and they characterized the oriented graphs which satisfy the equality in this paper we aim to establish an upper bound for skew energy of an oriented graph in terms of its skew rank and maximum vertex degree it is proved that e s g r s g for an arbitrary oriented graph g with maximum degree and the upper bound is attained if and only if g is the disjoint union of r s g 2 copies of k together with some isolated vertices where the orientation is switching equivalent to the elementary orientation of k which assigns all edges the same direction from a chromatic set to another one

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

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

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

ordering of oriented unicyclic graphs by skew energies ordering of the oriented unicyclic graphs with n vertices according to their minimal skew energies is investigated we derive the first 22 oriented unicyclic graphs for n 1784 and a series of oriented unicyclic graphs for 3 n 1783

bicyclic oriented graphs with skew rank 6 let g be an oriented graph and s g be its skew adjacency matrix the skew rank of g denoted by sr g is the rank of s g in this paper we characterize all the bicyclic oriented graphs with skew rank 6 let g be a bicyclic oriented graph with pendant vertices but no pendant twins if s r g 6 then 6 v g 10

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

5 regular oriented graphs with optimum skew energy abstract let g be a simple undirected graph and g be the corresponding oriented graph of g with the orientation the skew energy of g denoted by e s g is defined as the sum of the singular values of the skew adjacency matrix s g in 2010 adiga et al certified that s g n where is the maximum degree of g of order n it has been shown that every 5 regular oriented graph with optimum skew energy has even neighborhood property that is each pair of neighborhoods of a graph have even number of common vertices in this paper we characterize all connected 5 regular graphs of order n with this property moreover we determine all connected 5 regular oriented graphs of order n with maximum skew energy

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

odd triperfect numbers we prove that an odd triperfect number has at least ten distinct prime factors 1 a positive number n is called a triperfect number if a n 3n where a n is the sum of the positive divisors of n six even triperfect numbers are known 2 4 5 7 19 31 151 2 3 11 43 127 29 3 11 31 28 5 7 19 37 73 25 3 7 23 3 5 however the existence of an odd triperfect ot number is an open question mcdaniel 1 and cohen 2 proved that an ot number has at least nine distinct prime factors and beck and najar 3 showed that it exceeds 1050 in this paper using the technique of 4 we prove theorem if n is ot n has at least ten distinct prime factors 2 throughout this paper we let

skew rank of an oriented graph in terms of matching number abstract an oriented graph g is a digraph without loops and multiple arcs where g is called the underlying graph of g let s g denote the skew adjacency matrix of g the rank of s g is called the skew rank of g denoted by sr g which is even since s g is skew symmetric li and yu 2015 12 proved that the skew rank of an oriented unicyclic graph g is either 2 m g 2 or 2 m g where m g denotes the matching number of g in this paper we extend this result to general cases it is proved that the skew rank of an oriented connected graph g is an even integer satisfying 2 m g 2 g sr g 2 m g where g e g v g 1 is the number of fundamental cycles also called the first betti number besides the oriented graphs satisfying sr g 2 m g 2 g are characterized definitely

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

identification of nonminimum phase systems using higher order statistics a method is presented for identification of linear time variant nonminimum phase systems when only output data are available the input sequence need not be independent but it must be non gaussian with some special properties described in the test the authors model a finite dimensional system as an arma autoregressive moving average rational function of known orders but the special cases of ar ma and all pass models are also considered to estimate the parameters of their model the authors utilize both second and higher order statistics of the output which may be contaminated by additive zero mean gaussian white noise of unknown variance the parameter estimators obtained are proved under mild conditions to be consistent simulations verify the performance of the proposed method in the case of relatively low signal to noise ratios and when there is a model order mismatch

direct estimation of blind zero forcing equalizers based on second order statistics most existing zero forcing equalization algorithms rely either on higher than second order statistics or on partial or complete channel identification we describe methods for computing fractionally spaced zero forcing blind equalizers with arbitrary delay directly from second order statistics of the observations without channel identification we first develop a batch type algorithm then adaptive algorithms are obtained by linear prediction and gradient descent optimization our adaptive algorithms do not require channel order estimation nor rank estimation compared with other second order statistics based approaches ours do not require channel identification at all on the other hand compared with the cma type algorithms ours use only second order statistics thus no local convergence problem exists and faster convergence can be achieved simulations show that our algorithms outperform most typical existing algorithms

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

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

on upper bounds for the energy of digraphs abstract the energy of a digraph d is defined as e d i 1 n re z i where z 1 z 2 z n are the possibly complex eigenvalues of d in this paper we first give an improved lower bound on the spectral radius of the digraph d using this result we obtain a new upper bound on the energy e d and characterize some extreme digraphs which attain this upper bound this result theoretically improves and generalizes some known results

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

prediction error method for second order blind identification blind channel identification methods based on the oversampled channel output are a problem of current theoretical and practical interest in this paper we introduce a second order blind identification technique based on a linear prediction approach in contrast to eigenstructure based methods it will be shown that the linear prediction error method is robust to order overdetermination an asymptotic performance analysis of the proposed estimation method is carried out consistency and asymptotic normality of the estimates is established a closed form expression for the asymptotic covariance of the estimates is given numerical simulations and investigations are finally presented to demonstrate the potential and the robustness of the proposed method

spectral norm of oriented graphs abstract we show that the spectral radius of a graph is an upper bound on the spectral norm of any of its orientations we also show that equality is attained only when the orientation is sink source as an application we find the extremal values of the spectral norm over the set of oriented trees oriented unicyclic graphs and connected digraphs with n vertices and n arcs

oriented unicyclic graphs with the first n 92 largest skew energies abstract let g be an oriented graph obtained by assigning an orientation to the edge set of a simple undirected graph g such that g becomes a directed graph let s g be the skew adjacency matrix of g the skew energy of g is defined as the sum of the absolute values of all eigenvalues of s g in this paper we provide a new method to compare the skew energies of two oriented graphs whose skew characteristic polynomials satisfy a given recurrence relation and determine the oriented unicyclic graphs of order n with the first n 9 2 largest skew energies for n 31

blind fractionally spaced equalization of noisy fir channels direct and adaptive solutions blind fractionally spaced equalizers reduce intersymbol interference using second order statistics without the need for training sequences methods for finding fir zero forcing blind equalizers directly from the observations are described and adaptive versions are developed in contrast most current methods require channel estimation as a first step to estimating the equalizer the direct methods can be zero forcing minimum mean square error or even minimum mean square error mmse within the class of zero forcing equalizers performance of the proposed methods and comparisons with existing approaches are shown for a variety of channels including an empirically measured digital microwave channel

the skew energy of tournaments abstract let s be the skew adjacency matrix of a digraph the skew energy of s denoted by e s is the sum of the absolute values of the all eigenvalues of s in this paper for skew adjacency matrices of tournaments of order n we will give lower and upper bounds of the skew energy we show that our lower bound is tight in the sense that it coincides with the minimum of e s of all skew adjacency matrices of tournaments of order n in addition we will introduce the skew energy which is a generalization of the skew energy

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

the characteristic polynomial and the matchings polynomial of a weighted oriented graph abstract let g be a weighted oriented graph with skew adjacency matrix s g then g is usually referred as the weighted oriented graph associated to s g denote by g the characteristic polynomial of the weighted oriented graph g which is defined as g det i n s g i 0 n a i g n i in this paper we begin by interpreting all the coefficients of the characteristic polynomial of an arbitrary real skew symmetric matrix in terms of its associated oriented weighted graph then we establish recurrences for the characteristic polynomial and deduce a formula on the matchings polynomial of an arbitrary weighted graph in addition some miscellaneous results concerning the number of perfect matchings and the determinant of the skew adjacency matrix of an unweighted oriented graph are given

lower bounds of the skew spectral radii and skew energy of oriented graphs abstract let g be a graph with maximum degree and let g be an oriented graph of g with skew adjacency matrix s g the skew spectral radius s g of g is defined as the spectral radius of s g the skew spectral radius has been studied but only few results about its lower bound are known this paper determines some lower bounds of the skew spectral radius and then studies the oriented graphs whose skew spectral radii attain the lower bound moreover we apply the skew spectral radius to the skew energy of oriented graphs which is defined as the sum of the norms of all the eigenvalues of s g and denoted by e s g as a result we obtain some lower bounds of the skew energy which improve the known lower bound obtained by adiga 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

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 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

bicyclic oriented graphs with skew rank 2 or 4 the skew rank of oriented graph g denoted by sr g is the rank of the skew adjacency matrix of g the skew rank is even since the skew adjacency matrix is skew symmetric in this paper we characterize the bicyclic oriented graphs with skew rank 2 or 4

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

a lower bound for odd triperfects if n is odd and a n 3n then n exceeds 1050 a natural number n is called triperfect if a n 3n six triperfects are known all even the existence of an odd triperfect is an open question kanold 11 showed that an odd triperfect a has at least nine distinct prime factors b is a square and c exceeds 1020 the purpose of this note is to improve the last result we prove theorem if n is odd triperfect then n exceeds 1050 the inequality a pa pa 3 thus lemma if n is triperfect and n 6 1 then n exceeds 10 and has at least 32 distinct prime factors assume n an odd triperfect and 3 n since a n 3n even powers of prime factors of a n divide n up to possibly powers of 3 this caution is necessary since if 3a n then 3a 1 i a n if pa 11 n then paa pa i n again up to power of 3 as 354 5 1025 it suffices to examine n assuming 3 fl n t even 2 t 54 we construct all sequences of primes p 3p pi p2 and of even naturals n t no n1 n2 so that a pi is the largest prime divisor of a p and b for each i ni 2 4 6 the sequences are extended until either a the product of known factors of n exceeds 105 b powers of 3 among the a s exceed t 1 or c n is shown to be triperfect for the most part complete prime factorizations of the a s were calculated although this was not always necessary of the 221 cases that occurred four were of type b and the remainder of type a no triperfects were found this proves the theorem from known results on perfect and multiperfect numbers we have corollary if n is odd multiperfect the n exceeds 105 factorizations were computer generated the authors acknowledge with gratitude the assistance of dr jacob gerlach department of mathematics university of wisconsin whitewater and of professor d h lehmer department of mathematics university of california berkeley the authors also wish to acknowledge the valuable suggestions of the referee received january 12 1981 revised june 22 1981 1980 mathematics subject classification primary ioa20

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

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

ill convergence of godard blind equalizers in data communication systems the existence of stable undesirable equilibria for the godard algorithm is demonstrated through a simple autoregressive ar channel model these undesirable equilibria correspond to local but nonglobal minima of the underlying mean cost function and thus permit the ill convergence of the godard algorithms which are stochastic gradient descent in nature simulation results confirm predicted misbehavior for channel input of constant modulus it is shown that attaining the global minimum of the mean cost necessarily implies correct equalization a criterion is also presented for allowing a decision at the equalizer as to whether a global or nonglobal minimum has been reached

blind sequence detection without channel estimation this paper proposes a new blind sequence estimation method for single input single output siso systems utilizing an optimal trellis search which is performed by a channel independent viterbi algorithm civa in contrast to the traditional viterbi algorithm that requires accurate channel estimation civa does not require channel coefficients instead the metrics are calculated from a bank of test vectors designed off line the proposed algorithm has outstanding performance under most of the channel conditions specifically it does not suffer from ill conditioned channels in addition it does not depend on channel correlation estimation and therefore has fast convergence simulations demonstrate its superior performance over even most training based equalization algorithms

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

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

upper bound for the trace norm of the laplacian matrix of a digraph and normally regular digraphs abstract the trace norm of m m n c is defined as m k 1 n k where 1 2 n 0 are the singular values of m i e the square roots of the eigenvalues of m m we are particularly interested in the trace norm l d a n i n where l d is the laplacian matrix of a digraph d with n vertices and a arcs and i n is the n n identity matrix when d g is a graph with n vertices and m edges then l d a n i n l g 2 m n i n l e g the laplacian energy of g introduced by gutman and zhou in 2006 we show that for a digraph d with n vertices and a arcs l d a n i n n a a 2 n i 1 n d i 2 where d 1 d n are the outer degrees of the vertices of d moreover the digraphs where this bound is attained are special classes of normally regular digraphs studied by jorgensen in 2015 6 finally we construct normally regular digraphs where the equality is attained

graph energy change due to edge deletion the energy of a graph is the sum of the singular values of its adjacency matrix we are interested in how the energy of a graph changes when edges are deleted examples show that all cases are possible increased decreased unchanged our goal is to find possible graph theoretical descriptions and to provide an infinite family of graphs for each case the main tool is a singular value inequality for complementary submatrices and its equality case

edge addition singular values and energy of graphs and matrices the energy of a graph matrix is the sum of the absolute values of its eigenvalues we investigate the result of duplicating removing an edge to the energy of a graph we also deal with the problem that which graphs g have the property that if the edges of g are covered by some subgraphs then the energy of g does not exceed the sum of the subgraphs energies the problems are addressed in the general setting of energy of matrices which leads us to consider the singular values too among the other results it is shown that the energy of a complete multipartite graph increases if a new edge added or an old edge is deleted

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

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

blind channel identification subspace tracking method without rank estimation subspace ss methods are an effective approach for blind channel identification however these methods also have two major disadvantages 1 they require accurate channel length estimation and or rank estimation of the correlation matrix which is difficult with noisy channels and 2 they require a large amount of computation for the singular value decomposition svd which makes it inconvenient for adaptive implementation although many adaptive subspace tracking algorithms can be applied the computational complexity is still o m sup 3 where m is the data vector length in this paper we introduce new recursive subspace algorithms using ulv updating and successive cancellation techniques the new algorithms do not need to estimate the rank of the correlation matrix furthermore the channel length can be overestimated initially and be recovered at the end by a successive cancellation procedure which leads to more convenient implementations the adaptive algorithm has computations of o m sup 2 in each recursion the new methods can be applied to either the single user or the multiuser cases simulations demonstrate their good performance

on oriented graphs whose skew spectral radii do not exceed 2 abstract let s g be the skew adjacency matrix of the oriented graph g of order n and 1 2 n be all eigenvalues of s g the skew spectral radius s g of g is defined as max 1 2 n in this paper we investigate oriented graphs whose skew spectral radii do not exceed 2

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

bicyclic oriented graphs with the second largest skew energy abstract let g be an oriented graph and p s g x det x i s g be the skew characteristic polynomial of its skew adjacency matrix s g the skew energy of g is defined to be the sum of the absolute values of eigenvalues of s g in this paper we firstly find a novel relation between the coefficients of skew characteristic polynomial of unicyclic graph with the third largest skew energy and bicyclic oriented graphs and then use it along with other techniques to characterize the bicyclic oriented graphs with the second largest skew energy

applications of a theorem by ky fan in the theory of graph energy abstract the energy of a graph g is equal to the sum of the absolute values of the eigenvalues of g which in turn is equal to the sum of the singular values of the adjacency matrix of g let x y and z be matrices such that x y z the ky fan theorem establishes an inequality between the sum of the singular values of z and the sum of the sum of the singular values of x and y this theorem is applied in the theory of graph energy resulting in several new inequalities as well as new proofs of some earlier known inequalities

on the average energy of circulant graphs we obtain upper and lower bounds on the average energy of circulant graphs with n vertices and regularity d the average is taken over all representations of such graphs by circulant adjacency matrices

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