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

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

numerical solution of nonlinear quasi stationary magnetic fields the problem of quasi stationary nonlinear magnetic fields is formulated in a variational form a galerkin finite element solution is constructed discretization in time is carried out by means of one and two step linear a stable schemes the resulting system of nonlinear equations is solved by newton s method a numerical example is given

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

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

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

graphs for small multiprocessor interconnection networks abstract let d be the diameter of a graph g and let 1 be the largest eigenvalue of its 0 1 adjacency matrix we give a proof of the fact that there are exactly 69 non trivial connected graphs with d 1 1 9 these 69 graphs all have up to 10 vertices and were recently found to be suitable models for small multiprocessor interconnection networks we also examine the suitability of integral graphs to model multiprocessor interconnection networks especially with respect to the load balancing problem in addition we classify integral graphs with small values of d 1 1 in connection with the load balancing problem for multiprocessor systems

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

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

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

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

on graphs whose energy exceeds the number of vertices abstract let g be a graph on n vertices and let 1 2 n be the eigenvalues of a 0 1 adjacency matrix of g the energy of g is e i 1 n i we characterize several classes of graphs for which e n

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

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

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

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

the thermodynamic limit of quantum coulomb systems part i general theory abstract this article is the first in a series dealing with the thermodynamic properties of quantum coulomb systems in this first part we consider a general real valued function e defined on all bounded open sets of r 3 our aim is to give sufficient conditions such that e has a thermodynamic limit this means that the limit e n n 1 exists for all regular enough sequence n with growing volume n and is independent of the considered sequence the sufficient conditions presented in our work all have a clear physical interpretation in the next paper we show that the free energies of many different quantum coulomb systems satisfy these assumptions hence have a thermodynamic limit

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

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 energy of unitary cayley graphs abstract a graph g of order n is called hyperenergetic if e g 2 n 2 where e g denotes the energy of g the unitary cayley graph x n has vertex set z n 0 1 2 n 1 and vertices a and b are adjacent if gcd a b n 1 these graphs have integral spectrum and play an important role in modeling quantum spin networks supporting the perfect state transfer we show that the unitary cayley graph x n is hyperenergetic if and only if n has at least two prime factors greater than 2 or at least three distinct prime factors in addition we calculate the energy of the complement of unitary cayley graph and prove that x n is hyperenergetic if and only if n has at least two distinct prime factors and n 2 p where p is a prime number by extending this approach for every fixed k n we construct families of k hyperenergetic non cospectral integral circulant n vertex graphs with equal energy

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

strongly regular graphs with maximal energy the energy of a graph is the sum of the absolute values of the eigenvalues of its adjacency matrix koolen and moulton have proved that the energy of a graph on n vertices is at most n 1 n 2 and that equality holds if and only if the graph is strongly regular with parameters n n n 2 n 2 n 4 n 2 n 4 such graphs are equivalent to a certain type of hadamard matrices here we survey constructions of these hadamard matrices and the related strongly regular graphs

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

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

upper bounds to the ground state energies of the one and two component charged bose gases we prove upper bounds on the ground state energies of the one and two component charged bose gases the upper bound for the one component gas agrees with the high density asymptotic formula proposed by l foldy in 1961 the upper bound for the two component gas agrees in the large particle number limit with the asymptotic formula conjectured by f dyson in 1967 matching asymptotic lower bounds for these systems were proved in references 10 and 11 the formulas of foldy and dyson which are based on bogolubov s pairing theory have thus been validated

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

on the chromatic number of integral circulant graphs integral circulant graphs are a generalization of unitary cayley graphs recently studied by klotz and sander the integral circulant graph x n d has vertices 0 1 n 1 and two vertices a and b are adjacent iff gcd x y n d where d d d n 1 d 3 we construct a family of counterexamples using exhaustive search algorithm for graph coloring and disprove the conjecture in this case

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

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

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

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

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

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

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

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

diagonalization of bosonic quadratic hamiltonians by bogoliubov transformations abstract we provide general conditions for which bosonic quadratic hamiltonians on fock spaces can be diagonalized by bogoliubov transformations our results cover the case when quantum systems have infinite degrees of freedom and the associated one body kinetic and paring operators are unbounded our sufficient conditions are optimal in the sense that they become necessary when the relevant one body operators commute

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

finite element solution of maxwell s equations for hyperthermia treatment planning abstract a general galerkin formulation is given for the periodic form of maxwell s equations on a bounded inhomogeneous medium the formulation conserves energy exactly and its dispersion relation is studied solutions for test problems representative of hyperthermia applications show good agreement with analytic solutions a representative clinical example is shown

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

finite element variational crimes in parabolic elliptic problems the study of a completely discretized variational problem corresponding to a nonlinear second order parabolic elliptic initial boundary value problem with an initial value only inl 2 is presented the discretization in time is done by the euler backward method the discretization in space by the finite element method with linear functions on triangular elements the changes in domain are taken into account the convergence and the unconditional stability of the method is proved under the assumption that the boundary is piecewise of classc 3 and the forma v w has a potential and is strongly monotone and lipschitz continuous as a by product the existence and uniqueness theorem is obtained