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

projective techniques and functional integration for gauge theories a general framework for integration over certain infinite dimensional spaces is first developed using projective limits of a projective family of compact hausdorff spaces the procedure is then applied to gauge theories to carry out integration over the non linear infinite dimensional spaces of connections modulo gauge transformations this method of evaluating functional integrals can be used either in the euclidean path integral approach or the lorentzian canonical approach a number of measures discussed are diffeomorphism invariant and therefore of interest to the connection dynamics version of quantum general relativity the account is pedagogical in particular prior knowledge of projective techniques is not assumed

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

submodularity and the traveling salesman problem in this paper we investigate the relationship between traveling salesman tour lengths and submodular functions this work is motivated by the one warehouse multi retailer inventory distribution problem with traveling salesman tour vehicle routing costs our goal is to find a submodular function whose values are close to those of optimal tour lengths through a central warehouse and a group of retailers our work shows that a submodular approximation to traveling salesman tour lengths whose error is bounded by a constant does not exist however we present heuristics that have errors which grow slowly with the number of retailers for the traveling salesman problem in the euclidean plane furthermore we perform computational tests that show that our submodular approximations of traveling salesman tour lengths have smaller errors than our theoretical worst case analysis would lead us to believe

on the concavity of delivery games abstract delivery games introduced by hamers borm van de leensel and tijs in 1994 are combinatorial optimization games that arise from delivery problems closely related to the chinese postman problem cpp they showed that delivery games are not necessarily balanced for delivery problems corresponding to the class of bridge connected euler graphs they showed that the related games are balanced this paper focuses on the concavity property for delivery games a delivery game arising from a delivery model corresponding to a bridge connected euler graph need not be concave the main result will be that for delivery problems corresponding to the class of bridge connected cyclic graphs which is a subclass of the class of bridge connected euler graphs the related delivery games are concave further we discuss some extreme points of the core and the value for this class of concave delivery games

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 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

a strongly polynomial algorithm to solve combinatorial linear programs khachiyan and recently karmarkar gave polynomial algorithms to solve the linear programming problem these algorithms have a small theoretical drawback namely the number of arithmetic steps depends on the size of the input numbers we present a polynomial linear programming algorithm whose number of arithmetic steps depends only on the size of the numbers in the constraint matrix but is independent of the size of the numbers in the right hand side and objective vectors in particular it gives a polynomial algorithm for the minimum cost flow and multicommodity flow problems in which the number of arithmetic steps is independent of the size of the costs and capacities the algorithm makes use of an existing polynomial linear programming algorithm the problem of whether any algorithm has a running time that is independent even of the size of the numbers in the constraint matrix remains open

on some balanced totally balanced and submodular delivery games this paper studies a class of delivery problems associated with the chinese postman problem and a corresponding class of delivery games a delivery problem in this class is determined by a connected graph a cost function defined on its edges and a special chosen vertex in that graph which will be referred to as the post office it is assumed that the edges in the graph are owned by different individuals and the delivery game is concerned with the allocation of the traveling costs incurred by the server who starts at the post office and is expected to traverse all edges in the graph before returning to the post office a graph g is called chinese postman submodular or for short cp submodular cp totally balanced cp balanced respectively if for each delivery problem in which g is the underlying graph the associated delivery game is submodular totally balanced balanced respectively for undirected graphs we prove that cp submodular graphs and cp totally balanced graphs are weakly cyclic graphs and conversely an undirected graph is shown to be cp balanced if and only if it is a weakly euler graph for directed graphs cp submodular graphs can be characterized by directed weakly cyclic graphs further it is proven that any strongly connected directed graph is cp balanced for mixed graphs it is shown that a graph is cp submodular if and only if it is a mixed weakly cyclic graph finally we note that undirected directed and mixed weakly cyclic graphs can be recognized in linear time

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

some results concerning the representation theory of the algebra underlying loop quantum gravity important characteristics of the loop approach to quantum gravity are a specific choice of the algebra a of kinematical observables and of a representation of a on a measure space over the space of generalized connections this representation is singled out by its elegance and diffeomorphism covariance recently in the context of the quest for semiclassical states states of the theory in which the quantum gravitational field is close to some classical geometry it was realized that it might also be worthwhile to study different representations of the algebra a the content of the present work is the observation that under some mild assumptions the mathematical structure of representations of a can be analyzed rather effortlessly to a certain extent each representation can be labeled by sets of functions and measures on the space of generalized connections that fulfill certain conditions

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

cost allocation for a tree network with heterogeneous customers we analyze a cost allocation problem which could naturally arise from a situation wherein a tree networkt n 0 e serving heterogeneous customers has to be constructed the customers located atn require some service from a central supplier located at vertex 0 the customers have heterogeneous preferences for the level or quality of service received from the central supplier we formulate the above cost allocation problem as a cooperative game referred to as anextended tree game the extended tree game is a proper extension of megiddo s 1978 tree game wherein all the customers have identical preferences regarding the level of service received we prove that an extended tree game is convex and we show that its shapley value can be computed in p n time wherep is the number of distinct preference levels we further provide a complete facial description of the core polytope of an extended tree game and demonstrate that even when there are only two classes of customers the number of nonredundant core constraints could be exponential in n nevertheless we are able to construct an p n algorithm to check the core membership of an arbitrary cost allocation which can be used to construct an p n 3 combinatorial algorithm to compute the nucleolus of an extended tree game finally we show that the complements of the facet defining coalitions for the core are all connected in an auxiliary tree graph with node setn

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

uniqueness of diffeomorphism invariant states on holonomy flux algebras loop quantum gravity is an approach to quantum gravity that starts from the hamiltonian formulation in terms of a connection and its canonical conjugate quantization proceeds in the spirit of dirac first one defines an algebra of basic kinematical observables and represents it through operators on a suitable hilbert space in a second step one implements the constraints the main result of the paper concerns the representation theory of the kinematical algebra we show that there is only one cyclic representation invariant under spatial diffeomorphisms while this result is particularly important for loop quantum gravity we are rather general the precise definition of the abstract algebra of the basic kinematical observables we give could be used for any theory in which the configuration variable is a connection with a compact structure group the variables are constructed from the holonomy map and from the fluxes of the momentum conjugate to the connection the uniqueness result is relevant for any such theory invariant under spatial diffeomorphisms or being a part of a diffeomorphism invariant theory

minimizing the laplacian eigenvalues for trees with given domination number abstract in this paper we study the laplacian spectral radius of trees on n vertices with domination number where n k k 2 is an integer and determine the extremal trees that attain the minimal laplacian spectral radius when 2 3 4

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

minimum energy on trees with k pendent vertices the energy of g denoted by e g is defined as the sum of the absolute values of the eigenvalues of g in this paper we characterize the tree with minimal energy among the trees with k pendent vertices

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

when do measures on the space of connections support the triad operators of loop quantum gravity in this work we investigate the question under what conditions hilbert spaces that are induced by measures on the space of generalized connections carry a representation of certain non abelian analogues of the electric flux we give the problem a precise mathematical formulation and start its investigation for the technically simple case of u 1 as gauge group we establish a number of no go theorems asserting that for certain classes of measures the flux operators can not be represented on the corresponding hilbert spaces the flux observables we consider play an important role in loop quantum gravity since they can be defined without recurse to a background geometry and they might also be of interest in the general context of quantization of non abelian gauge theories

minimal energies of trees with given parameters abstract the energy of a graph is the sum of the absolute values of the eigenvalues of its adjacency matrix let t k be the set of trees with given order k suppose that t t k and v 1 v 2 v k be the ordering vertex set of t we denote by t n 1 n 2 n k the graph obtained by attaching n i pendent vertices to vertex v i i 1 2 k of t respectively let t n k t n 1 n 2 n k t t k n 1 n 2 n k n k n i 1 i 1 2 k in this paper we determine the trees in t n k with the first and the second minimal energies as applications we can characterize the trees with the first and the second minimal energies among the set of trees with given domination number matching number independence number respectively

highway games on weakly cyclic graphs a highway problem is determined by a connected graph which provides all potential entry and exit vertices and all possible edges that can be constructed between vertices a cost function on the edges of the graph and a set of players each in need of constructing a connection between a specific entry and exit vertex mosquera 2007 introduce highway problems and the corresponding cooperative cost games called highway games to address the problem of fair allocation of the construction costs in case the underlying graph is a tree in this paper we study the concavity and the balancedness of highway games on weakly cyclic graphs a graph g is called highway game concave if for each highway problem in which g is the underlying graph the corresponding highway game is concave we show that a graph is highway game concave if and only if it is weakly triangular moreover we prove that highway games on weakly cyclic graphs are balanced

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

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

monotonic stable solutions for minimum coloring games for the class of minimum coloring games introduced by deng et al math oper res 24 751 766 1999 we investigate the existence of population monotonic allocation schemes introduced by sprumont games econ behav 2 378 394 1990 we show that a minimum coloring game on a graph g has a population monotonic allocation scheme if and only if g is p 4 2k 2 free or equivalently if its complement graph bar g is quasi threshold moreover we provide a procedure that for these graphs always selects an integer population monotonic allocation scheme

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

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

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