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

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

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

new results on the energy of integral circulant graphs abstract circulant graphs are an important class of interconnection networks in parallel and distributed computing integral circulant graphs play an important role in modeling quantum spin networks supporting the perfect state transfer as well the integral circulant graph icg n d has the vertex set z n 0 1 2 n 1 and vertices a and b are adjacent if gcd a b n d where d d d n 1 d n these graphs are highly symmetric have integral spectra and some remarkable properties connecting chemical graph theory and number theory the energy of a graph was first defined by gutman as the sum of the absolute values of the eigenvalues of the adjacency matrix recently there was a vast research for the pairs and families of non cospectral graphs having equal energies following bapat and pati r b bapat s pati energy of a graph is never an odd integer bull kerala math assoc 1 2004 129 132 we characterize the energy of integral circulant graph modulo 4 furthermore we establish some general closed form expressions for the energy of integral circulant graphs and generalize some results from ilic a ilic the energy of unitary cayley graphs linear algebra appl 431 2009 1881 1889 we close the paper by proposing some open problems and characterizing extremal graphs with minimal energy among integral circulant graphs with n vertices provided n is even

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

subspace based prototyping and classification of chromosome images chromosomes are essential genomic information carriers chromosome classification constitutes an important part of routine clinical and cancer cytogenetics analysis cytogeneticists perform visual interpretation of banded chromosome images according to the diagrammatic models of various chromosome types known as the ideograms which mimic artists depiction of the chromosomes in this paper we present a subspace based approach for automated prototyping and classification of chromosome images we show that 1 prototype chromosome images can be quantitatively synthesized from a subspace to objectively represent the chromosome images of a given type or population and 2 the transformation coefficients or projected coordinate values of sample chromosomes in the subspace can be utilized as the extracted feature measurements for classification purposes we examine in particular the formation of three well known subspaces namely the ones derived from principal component analysis pca fisher s linear discriminant analysis and the discrete cosine transform dct these subspaces are implemented and evaluated for prototyping two dimensional 2 d images and for classification of both 2 d images and one dimensional profiles of chromosomes experimental results show that previously unseen prototype chromosome images of high visual quality can be synthesized using the proposed subspace based method and that pca and the dct significantly outperform the well known benchmark technique of weighted density distribution functions in classifying 2 d chromosome images

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

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

the dynamic stability of a simply supported beam with additional discrete elements the dynamic stability of a simply supported beam with additional discrete elements was investigated in the paper those elements were an elastic spring a concentrated mass and an undamped harmonic oscillator connected to the beam all the discrete elements could be mounted at any chosen position along the beam length the beam was axially loaded by a harmonic force the problem of dynamic stability was solved by applying the mode summation method the obtained mathieu equation allowed the influence of additional elements on the position of solutions on a stability chart to be analysed the analysis relied on testing the influence of individual discrete elements on the value of coefficient b in the mathieu equation the research carried out showed that both the concentrated mass and oscillator mass had a destabilising effect maximum in the middle position on the investigated system the rigidity of the support and the oscillator had an influence on an increase in the stability of the investigated system an increase in the loading force independently of the relation between the mass and rigidity of discrete elements had an influence on the increase in coefficient b in the mathieu equation the less stable system the considered beam is treated as a bernoulli euler beam in accordance with the small bending theory

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

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 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

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

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

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

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

more on borderenergetic graphs abstract the energy e g of a graph g is defined as the sum of the absolute values of the eigenvalues of its adjacency matrix if a graph g of order n has the same energy as the complete graph k n i e if e g 2 n 1 then g is said to be borderenergetic we obtain three asymptotically tight bounds on the edge number of borderenergetic graphs then by using disconnected regular graphs we construct connected non complete borderenergetic graphs

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

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

parametric vibrations of a restrained beam with an end mass under displacement excitation this paper is concerned with the stability and the steady state response of the main parametric resonance vibrations of a simply supported vertical beam the beam carries a concentrated mass and is restrained at one end and subjected to a periodic axial displacement excitation at the other end this system can be looked upon as a dynamic model of the vibrations of an engine valve mechanism non linear terms arising from moderately large curvatures longitudinal inertia of the beam elements and concentrated mass are included in the equation of motion by using the one mode approximation and applying galerkin s method the governing partial differential equation is reduced to a non linear ordinary differential equation with a periodic coefficient the boundaries of the main parametric instability region of the linearized equation are obtained the harmonic balance method is applied to solve the equation and an analytical expression for the dynamic response in the vicinity of the main parametric resonance is derived the effects of various parameters on the boundaries of the instability region and the dynamic response are investigated

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

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

assessment of lateral dynamic instability of columns under an arbitrary periodic axial load owing to parametric resonance abstract the paper presents a method for assessing the dynamic stability of simply supported columns with damping under arbitrary periodic axial loading owing to parametric resonance the equation for determining critical excitation frequencies causing lateral parametric instability of columns is derived using the bolotin s method a method of matrix transformation is developed to solve the ill condition problem in the even regions of dynamic instability of the column caused by inclusion of damping the accuracy of the proposed method is carefully verified by comparing the results with those in the literature new findings of this investigation are 1 the critical excitation parameter for inducing dynamic instability under arbitrary periodic loading increases with increasing damping ratio 2 in difference from columns under a simple harmonic load the width of instability regions of columns under an arbitrary periodic load does not necessarily decrease as the number of region increases so the instability regions can cover most of the parametric plane and thus the dynamic instability is more likely to occur 3 under an arbitrary periodic axial load the first region of instability of columns may not be the critical one and the number of harmonic waves adopted in the computation is important for the location of the most critical region of instability and the distribution of instability regions in the parametric plane and 4 the dimension of determinant in the equation of boundary frequencies rather than the number of harmonic waves is responsible for computation efficiency and the sufficient large dimension of determinant should be retained to achieve the computational accuracy

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

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

simultaneous combination principal parametric and internal resonances in a slender beam with a lumped mass three mode interactions abstract the non linear dynamics of a parametrically base excited slender beam carrying a lumped mass with an excitation frequency nearly equal to twice the second mode natural frequency is examined the mass and the position of the attached element are so adjusted that the system exhibits 1 3 5 internal resonance by virtue of which a combination resonance of type 1 3 occurs simultaneously in the system the method of multiple scales mms is applied to reduce the second order temporal differential equations of motion of the system to a set of first order differential equations which is used to study the steady state periodic quasi periodic and chaotic responses of the system for different control parameters namely frequency and amplitude of base excitation damping internal detuning etc the results of perturbation analysis are compared with those obtained by numerically integrating the original temporal equations of motion poincare section and lyapunov exponents are used to characterize the chaotic responses

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

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

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

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

the response of a single degree of freedom system with quadratic and cubic non linearities to a fundamental parametric resonance abstract the response of a one degree of freedom system with quadratic and cubic non linearities to a principal parametric resonance is investigated the method of multiple scales is used to determine the equations that describe to second order the modulation of the amplitude and phase with time about one of the foci these equations are used to determine the fixed points and their stability the perturbation results are verified by integrating the governing equation with use of a digital computer and an analogue computer for small excitation amplitudes the analytical results are in excellent agreement with the numerical solutions the large amplitude responses are investigated by using both a digital and an analogue computer the cases of single and double well potentials are investigated systems with double well potentials exhibit complicated dynamic behaviors including period multiplying and demultiplying bifurcations and chaos in some cases a bifurcated response coexists with another periodic attractor and a chaotic attractor coexists with a periodic attractor long time histories phase planes poincare maps fractal basin maps and spectra of the response are presented a bifurcation diagram of many solutions in the excitation amplitude excitation frequency plane is also presented

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 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 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

automatic classification of chromosomes by means of quadratically asymmetric statistical distributions abstract we use quadratically asymmetric distributions as introduced in ritter 1 2 as statistical models of human chromosomes for automatic constrained bayesian classification into their 24 classes these distributions are able to reflect asymmetries in the data moreover we design algorithms for constrained classification of cells with missing and extra chromosomes trisomies applied to the edinburgh features of the large copenhagen data set cpr the best classifier reported here reduces the cross validation error rate from 2 7 classical normal model to 1 2 with respect to chromosomes on the average five out of six cells are completely correctly classified

on the normalized laplacian energy and general randic index r 1 of graphs in this paper we consider the energy of a simple graph with respect to its normalized laplacian eigenvalues which we call the l energy over graphs of order n that contain no isolated vertices we characterize the graphs with minimal l energy of 2 and maximal l energy of 2 n 2 we provide upper and lower bounds for l energy based on its general randic index r 1 g we highlight known results for r 1 g most of which assume g is a tree we extend an upper bound of r 1 g known for trees to connected graphs we provide bounds on the l energy in terms of other parameters one of which is the energy with respect to the adjacency matrix finally we discuss the maximum change of l energy and r 1 g upon edge deletion

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

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

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

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

lower 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 n are the eigenvalues of d in this article we find lower bounds for the energy of digraphs in terms of the number of closed walks of length 2 extending in this way the result obtained by caporossi et al g caporossi d cvetkovic i gutman p hansen variable neighborhood search for extremal graphs 2 finding graphs with extremal energy j chem inf comput sci 39 1999 984 996 2 m e g for all graphs g with m edges also we study digraphs with three eigenvalues

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

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

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

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

on the laplacian coefficients of graphs under some transformations abstract we generalize three approaches on graph transformations respectively from stevanovic and ilic 2009 16 and tan 2011 19 we also generalize an approach of graph transformations on the spectral radius of adjacency matrix into the laplacian coefficients of graphs from li and feng 1979 26 moreover we determine the unique tree having the third maximal laplacian coefficients among all n vertex trees

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

using dominant points and variants for profile extraction from chromosomes abstract the most accurate methods for automatic analysis of chromosomes under a light microscope today extract numerical features from band pattern profiles along their longitudinal axes the construction of a reliable axis is a crucial step in this process we propose a new way based on the dominant points of the contour and cubic splines the dominant points serve as candidates for the tips of the chromosome ambiguities are dissolved by the recently developed method of variants for object identification a voronoi diagram decomposes the chromosome in slices for profile extraction the method improves the currently best classification results significantly yielding a test set error rate of 0 61

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

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

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

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

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

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

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

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

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

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

improved bounds for the laplacian energy of bethe trees abstract for k d 2 a bethe tree is a rooted tree with k levels which the root vertex has degree d the vertices from level 2 to k 1 have degree d 1 and the vertices at the level k are pendent vertices so et al using a theorem by ky fan have obtained both upper and lower bounds for the laplacian energy of bipartite graphs we shall employ the above mentioned theorem to obtain new and improved bounds for the laplacian energy in the case of bethe trees

the laplacian energy of random graphs abstract gutman et al introduced the concepts of energy e g and laplacian energy e l g for a simple graph g and furthermore they proposed a conjecture that for every graph g e g is not more than e l g unfortunately the conjecture turns out to be incorrect since liu et al and stevanovic et al constructed counterexamples however so et al verified the conjecture for bipartite graphs in the present paper we obtain for a random graph the lower and upper bounds of the laplacian energy and show that the conjecture is true for almost all graphs