bicyclic graphs with small positive index of inertia abstract the positive index of inertia of a graph g denoted by i g is the number of the positive eigenvalues of the adjacency matrix of g in this paper we investigate the minimal positive index of inertia among all bicyclic graphs of order n with pendant vertices and characterize the bicyclic graphs with positive index 1 or 2 among all bicyclic graphs of order n

structural similarity of directed universal hierarchical graphs a low computational complexity approach in the present paper we mainly introduce an efficient approach to measure the structural similarity of so called directed universal hierarchical graphs we want to underline that directed universal hierarchical graphs can be obtained from generalized trees which are already introduced in order to classify these graphs we state our novel graph similarity method as a main result we notice that our novel algorithm has low computational complexity

a polynomial algorithm of edge neighbor scattering number of trees abstract the edge neighbor scattering number ens is an alternative invulnerability measure of networks such as the vertices represent spies or virus carriers let g v e be a graph and e be any edge in g the open edge neighborhood of e is n e f e g f e e and f are adjacent and the closed edge neighborhood of e is n e n e e an edge e in g is said to be subverted when n e is deleted from g an edge set x e g is called an edge subversion strategy of g if each of the edges in x has been subverted from g the survival subgraph is denoted by g x an edge subversion strategy x is called an edge cut strategy of g if the survival subgraph g x is disconnected or is a single vertex or is the ens of a graph g is defined as e n s g max x e g g x x where x is any edge cut strategy of g g x is the number of the components of g x it is proved that the problem of computing the ens of a graph is np complete in this paper we give a polynomial algorithm of ens of trees

information processing in complex networks graph entropy and information functionals this paper introduces a general framework for defining the entropy of a graph our definition is based on a local information graph and on information functionals derived from the topological structure of a given graph more precisely an information functional quantifies structural information of a graph based on a derived probability distribution such a probability distribution leads directly to an entropy of a graph then the structural information content of a graph will be is interpreted and defined as the derived graph entropy another major contribution of this paper is the investigation of relationships between graph entropies in addition to this we provide numerical results demonstrating not only the feasibility of our method which has polynomial time complexity but also its usefulness with regard to practical applications aiming to an understanding of information processing in complex networks

unicyclic graphs with large energy we study the energy i e the sum of the absolute values of all eigenvalues of so called tadpole graphs which are obtained by joining a vertex of a cycle to one of the ends of a path by means of the coulson integral formula and careful estimation of the resulting integrals we prove two conjectures on the largest and second largest energy of a unicyclic graph due to caporossi cvetkovic gutman and hansen and gutman furtula and hua respectively moreover we characterise the non bipartite unicyclic graphs whose energy is largest

a comparative analysis of new graph distance measures and graph edit distance the problem of determining the structural similarity or distance between graphs got considerable attention during the last decades consequently various similarity and distance measures for networks have already been investigated dehmer etal studied a new distance measure for graphs which is based on topological indices an interesting problem is to compare our class of measures with the well known graph edit distance which has been studied extensively and often serves as a benchmark measure but note that ged is generally np hard in this paper we compare the edit distance with our graph distance measures in order to do so we use some well known topological indices such as graph energy wiener index randi index and graph entropy by using several special graph classes numerical results reveal that the graph distance measure based on graph energy approximates ged well this fact could trigger an important research direction for studying graph edit distance and other comparative network measures

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

inertia of complex unit gain graphs let t z c z 1 be a subgroup of the multiplicative group of all nonzero complex numbers c a t gain graph is a triple g t consisting of a graph g v e the circle group t and a gain function e t such that e i j e j i 1 e j i the adjacency matrix a of the t gain graph g of order n is an n n complex matrix aij where a i j e i j if v i is adjacent to v j 0 otherwise evidently this matrix is hermitian the inertia of is defined to be the triple i n i i i 0 where i i i 0 are numbers of the positive negative and zero eigenvalues of a including multiplicities respectively in this paper we investigate some properties of inertia of t gain graph

instability of the cauchy kovalevskaya solution for a class of nonlinear systems we prove that in any c infty neighborhood of an analytic cauchy datum there exists a smooth function such that the corresponding initial value problem does not have any classical solution for a class of first order nonlinear systems we use a method initiated by g m etivier for elliptic systems based on the representation of solutions and on the fbi transform in our case the system can be hyperbolic at initial time but the characteristic roots leave the real line at positive times

a note on extremal trees with degree conditions abstract a fundamental question in the study of graph invariants asks for the extremal structures under certain constraints that maximize or minimize a graph invariant in this note we summarize some recent work on the extremal trees of distance based and degree based graph invariants under various degree conditions we note that many of such extremal structures turned out to be identical for different but similar invariants such common extremal structures are investigated through the greedy trees and majorization between degree sequences we show that many of the known extremal results can be obtained through this line of arguments we also introduce some new extremal results as immediate consequences

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

a torelli theorem for simply connected elliptic surfaces with a section and p_g geqslant 2 0 the purpose of this note is to announce some recent results on the period mapping for simply connected elliptic surfaces with a section and p 2 among other things we are able to prove the period mapping for such surfaces has degree one that is to say it is generically injective in the terminology of c g this is called the weak global torelli theorem in order to make a precise statement of our results we must introduce some notation

on graph entropy measures based on the number of independent sets and matchings abstract in this paper we consider the graph entropy measures based on the number of independent sets and matchings the reason to study these measures relates to the fact that the independent set and matching problem is computationally demanding however these features can be calculated for smaller networks in case one can establish efficient estimations those measures may be also used for larger graphs so we establish some upper and lower bounds as well as some information inequalities for these information theoretic quantities in order to give further evidence we also generate numerical results to study these measures such as list the extremal graphs for these entropies those results reveal the two entropies possess some new features

degree based entropies of networks revisited studies on the information content of graphs and networks have been initiated in the late fifties based on the seminal work due to shannon and rashevsky various graph parameters have been used for the construction of entropy based measures to characterize the structure of complex networks based on shannon s entropy in cao et al extremality of degree based graph entropies inform sci 278 2014 22 33 we studied graph entropies which are based on vertex degrees by using so called information functionals as a matter of fact there has been very little work to find their extremal values when considered shannon entropy based graph measures we pursue with this line of research by proving further extremal properties of the degree based graph entropies

second order randic index of fluoranthene type benzenoid systems fluoranthene type benzenoid system are polycyclic conjugated molecules consisting of two benzenoid fragments connected by two carbon carbon bonds so as to form a five membered ring fluoranthene type benzenoid system and its congeners are compounds which in view of their chemical and physical properties belong among benzenoid hydrocarbons let f denotes a fluoranthene type benzenoid system whose second order randic index is denoted by r2 f in this paper we give the expression of r2 f in terms of their inlet features and we find the minimal and maximal value of the second order randic index over the set of cata catacondensed fluoranthene type benzenoid systems and characterize their corresponding graphs

entropy bounds for dendrimers abstract many graph invariants have been used for the construction of entropy based measures to characterize the structure of complex networks when considering shannon entropy based graph measures there has been very little work to find their extremal values a reason for this might be the fact that shannon s entropy represents a multivariate function and all probability values are not equal to zero when considering graph entropies dehmer and kraus proved some extremal results for graph entropies which are based on information functionals and express some conjectures generated by numerical simulations to find extremal values of graph entropies dehmer and kraus discussed the extremal values of entropies for dendrimers in this paper we continue to study the extremal values of graph entropy for dendrimers which has most interesting applications in molecular structure networks and also in the pharmaceutical and biomedical area among all dendrimers with n vertices we obtain the extremal values of graph entropy based on different well known information functionals numerical experiments verifies our results

graph operations based on using distance based graph entropies abstract let g be a connected graph the eccentricity of vertex v is the maximum distance between v and other vertices of g in this paper we study a new version of graph entropy based on eccentricity of vertices of g in continuing we study this graph entropy for some classes of graph operations finally we compute the graph entropy of two classes of fullerene graphs

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

offdiagonal complexity a computationally quick complexity measure for graphs and networks abstract a vast variety of biological social and economical networks shows topologies drastically differing from random graphs yet the quantitative characterization remains unsatisfactory from a conceptual point of view motivated from the discussion of small scale free networks a biased link distribution entropy is defined which takes an extremum for a power law distribution this approach is extended to the node node link cross distribution whose nondiagonal elements characterize the graph structure beyond link distribution cluster coefficient and average path length from here a simple and computationally cheap complexity measure can be defined this offdiagonal complexity odc is proposed as a novel measure to characterize the complexity of an undirected graph or network while both for regular lattices and fully connected networks odc is zero it takes a moderately low value for a random graph and shows high values for apparently complex structures as scale free networks and hierarchical trees the odc approach is applied to the helicobacter pylori protein interaction network and randomly rewired surrogates

comparing large graphs efficiently by margins of feature vectors measuring the structural similarity of graphs is a challenging and outstanding problem most of the classical approaches of the so called exact graph matching methods are based on graph or subgraph isomorphic relations of the underlying graphs in contrast to these methods in this paper we introduce a novel approach to measure the structural similarity of directed and undirected graphs that is mainly based on margins of feature vectors representing graphs we introduce novel graph similarity and dissimilarity measures provide some properties and analyze their algorithmic complexity we find that the computational complexity of our measures is polynomial in the graph size and hence significantly better than classical methods from e g exact graph matching which are np complete numerically we provide some examples of our measure and compare the results with the well known graph edit distance

on the zeros of the partial hosoya polynomial of graphs abstract the partial hosoya polynomial or briefly the partial h polynomial can be used to construct the well known hosoya polynomial the ith coefficient of this polynomial defined for an arbitrary vertex u of a graph g is the number of vertices at distance i from u the aim of this paper is to determine the partial h polynomial of several well known graphs and then to investigate the location of their zeros to pursue we characterize the structure of graphs with the minimum and the maximum modulus of the zeros of partial h polynomial finally we define another graph polynomial of the partial h polynomial see 9 also we determine the unique positive root of this polynomial for particular graphs

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

the general connectivity indices of fluoranthene type benzenoid systems the general connectivity index r g of a graph g v e is defined as r g u v e g d u d v where u v is an edge of g du and dv denote the degrees of the vertices u and v respectively and 0 is an arbitrary real number in this paper we give the expressions for computing the general connectivity indices of fluoranthene type benzenoid systems and we determine the extremal values of r g in f benzenoid systems with h hexagons for some real number especially the extremal values of r g in cata catacondensed fluoranthene type benzenoid systems with h hexagons were studied

finding a shortest cycle in a subspace of the cycle space of a graph thomassen s 3 path condition shows that it is relatively easy for one to find a shortest cycle in a collection of cycles beyond a subspace of the cycle space of a connected graph and the real challenge is to find a shortest cycle contained in a given subspace of the cycle space of a graph in this article we investigate the shorter cycle structures in a given subspace of a graph and find a set of cycles in a given graph containing much information about short cycles we show that for a large range of subspaces of a graph satisfying a parity condition there exists a polynomial time algorithm to find a shortest cycle in these subspaces this makes a unified treatment of several famous algorithms finally we provide lower bounds of some types of short cycles in embedded graphs

relations between distance based and degree based topological indices let w sz pi and wp be respectively the wiener szeged pi and wiener polarity indices of a molecular graph g let m1 and m2 be the first and second zagreb indices of g we obtain relations between these classical distance and degree based topological indices

on the degeneracy of the randic entropy and related graph measures abstract numerous quantitative graph measures have been defined and applied in various disciplines such measures may be differentiated according to whether they are information theoretic or non information theoretic in this paper we examine an important property of randic entropy an information theoretic measure and examine some related graph measures based on random roots in particular we investigate the degeneracy of these structural graph measures and discuss numerical results finally we draw some conclusions about the measures applicability to deterministic and non deterministic networks

entropy measures for networks toward an information theory of complex topologies the quantification of the complexity of networks is today a fundamental problem in the physics of complex systems a possible roadmap to solve the problem is via extending key concepts of information theory to networks in this rapid communication we propose how to define the shannon entropy of a network ensemble and how it relates to the gibbs and von neumann entropies of network ensembles the quantities we introduce here will play a crucial role for the formulation of null models of networks through maximum entropy arguments and will contribute to inference problems emerging in the field of complex networks

bounds for degree based network entropies in this paper we continue studying degree based entropies for networks the quantities represent information theoretic network measures which are based on using information functionals involving vertex degrees we prove bounds for entropies which are based on information functionals using degree powers and come up with interrelations between different measures such interrelations are important to study connections between the measures required to understand the measures in depth

on the extremal values of general degree based graph entropies in this note we prove one part of the conjecture about upper and lower bounds of the degree based graph entropy ik t in the class of trees introduced in s cao m dehmer y shi extremality of degree based graph entropies information sciences 278 2014 22 33 using lagrange multipliers and jensen s inequality and disprove the other part by providing a family of counter examples our main result is the following the path pn is unique tree on n vertices that maximizes ik t for k 0 and the star sn is unique tree on n vertices that minimizes ik t for k 1

further results on digraphs with completely real laplacian spectra let n be the digraph with vertex set v n 1 2 n and arc set e l k k l k 2 n k k 1 k 1 n 1 and n i j the digraph obtained by deleting one arc i j from n where 1 j

complete solution to a problem on the maximal energy of unicyclic bipartite graphs abstract the energy of a simple graph g denoted by e g is defined as the sum of the absolute values of all eigenvalues of its adjacency matrix denote by c n the cycle and p n 6 the unicyclic graph obtained by connecting a vertex of c 6 with a leaf of p n 6 caporossi et al conjectured that the unicyclic graph with maximal energy is p n 6 for n 8 12 14 and n 16 in hou et al 2002 y hou i gutman c woo unicyclic graphs with maximal energy linear algebra appl 356 2002 27 36 the authors proved that e p n 6 is maximal within the class of the unicyclic bipartite n vertex graphs differing from c n and they also claimed that the energies of c n and p n 6 is quasi order incomparable and left this as an open problem in this paper by utilizing the coulson integral formula and some knowledge of real analysis especially by employing certain combinatorial techniques we show that the energy of p n 6 is greater than that of c n for n 8 12 14 and n 16 which completely solves this open problem and partially solves the above conjecture

counterexamples to conjectures on graph distance measures based on topological indexes in this paper we disprove three conjectures from dehmer et al 2014 on graph distance measures based on topological indices by providing explicit classes of trees that do not satisfy proposed inequalities the constructions are based on the families of trees that have the same wiener index graph energy or randic index but different degree sequences

the inertia of weighted unicyclic graphs let gw be a weighted graph the inertia of gw is the triple in gw i gw i gw i0 gw where i gw i gw i0 gw are the numbers of the positive negative and zero eigenvalues of the adjacency matrix a gw of gw including their multiplicities respectively i gw i gw are called the positive negative indices of inertia of gw respectively in this paper we present a lower bound for the positive negative indices of weighted unicyclic graphs of order n with fixed girth and characterize all weighted unicyclic graphs attaining this lower bound moreover we characterize the weighted unicyclic graphs of order n with two positive two negative and at least n 6 zero eigenvalues respectively

information theoretic description of networks we present a new information theoretic approach for network characterizations it is developed to describe the general type of networks with n nodes and l directed and weighted links i e it also works for the simpler undirected and unweighted networks the new information theoretic measures for network characterizations are based on a transmitter receiver analogy of effluxes and influxes based on these measures we classify networks as either complex or non complex and as either democracy or dictatorship networks directed networks in particular are furthermore classified as either information spreading and information collecting networks the complexity classification is based on the information theoretic network complexity measure medium articulation ma it is proven that special networks with a medium number of links l n1 5 show the theoretical maximum complexity ma logn 2 2 a network is complex if its ma is larger than the average ma of appropriately randomized networks ma mar a network is of the democracy type if its redundancy r h b a it is an information collecting network otherwise an information spreading network finally different real networks directed and undirected weighted and unweighted are classified according to our general scheme

a rudin carleson theorem for planar vector fields this paper generalizes the rudin carleson theorem for homogeneous solutions of locally solvable real analytic vector fields

comparisons of karci and shannon entropies and their effects on centrality of social networks abstract in order to measure the amount of different information in a system entropy concept can be used graph entropy measures nodes contribution to the entropy of the graph by this way the influential actors can be identified due to this case a new entropy based method was proposed to identify the influential actors karci entropy was applied to the social networks first time the alpha parameter allowed us to combine many different conditions together when measuring in the network the other important contribution of this paper is to predict the value of alpha parameter of karci entropy by using fuzzy logic after that karci and shannon entropies were compared based on experimental results moreover karci entropy was compared to traditional centrality measures if karci entropy definition is considered as a set of entropies shannon entropy can be regarded as an element of this set accordingly it can be concluded that karci entropy is superior to shannon entropy

connective steiner 3 eccentricity index and network similarity measure abstract for a set s v g in a network g the steiner distance dg s of s is the minimum size among all connected subnetworks whose vertex sets contain s the steiner k eccentricity k v of a vertex v of g is the maximum steiner distance among all k vertex set s which contains the vertex v i e e k v max d s s v g s k v s based on steiner k eccentricity the connective steiner k eccentricity index is introduced as a newly structural invariant some properties of the connective steiner 3 eccentricity index are investigated firstly we present an o n2 polynomial time algorithm to calculate the connective steiner 3 eccentricity index of trees secondly some optimal problems among some network classes are discussed as its application finally we consider the network similarity measure based on the connective steiner 3 eccentricity index by two different methods we study its advantages numerical results show that the measure based on the connective steiner 3 eccentricity index has more advantages than the ones based on other topological indices graph energy randic index the largest adjacent eigenvalue the largest laplacian eigenvalue

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

probabilistic inequalities for evaluating structural network measures proving interrelations between structural graph measures analytically has been intricate generally relations between structural graph measures describe the interplay between measures which turned out to be useful for better understanding the properties of such quantities the results which have been achieved so far are restricted to simple measures or special graph classes such as trees in this paper we introduce a probabilistic approach for establishing inequalities between quantitative network measures on random networks those inequalities between different graph measures lead to a deeper understanding of the mathematical apparatus and in particular to a discussion of quality aspects of structural graph measures which is a major contribution of this paper

a lower bound of revised szeged index of bicyclic graphs abstract the revised szeged index of a graph is defined as s z g e u v e n u e n 0 e 2 n v e n 0 e 2 where nu e and nv e are respectively the number of vertices of g lying closer to vertex u than to vertex v and the number of vertices of g lying closer to vertex v than to vertex u and n0 e is the number of vertices equidistant to u and v in the paper we identify the lower bound of revised szeged index among all bicyclic graphs and also characterize the extremal graphs that attain the lower bound

on rainbow domination numbers of graphs given a graph g and a set of t colors assume that we assign an arbitrary subset of these colors to each vertex of g if we require that each vertex to which an empty set is assigned has in its neighborhood all t colors then this assignment is called a t rainbow dominating function of the graph g the corresponding invariant c r t g which is the minimum sum of numbers of assigned colors over all vertices of g is called the t rainbow domination number of g in this paper bounds for the t rainbow domination number of an arbitrary graph for an arbitrary positive integer t are given the 3 rainbow domination numbers of several classes of graphs such as paths cycles and the generalized petersen graphs p n k are investigated the 3 rainbow domination number of p n 1 is determined and the upper bounds for p n 2 and p n 3 are provided by computer search we obtain that the upper bounds for p n 2 match their exact values for n 17

note on two generalizations of the randic index for a given graph g the well known randic index of g introduced by milan randic in 1975 is defined as r g u v e g d u d v 1 2 where the sum is taken over all edges uv and du denotes the degrees of u bollobas and erdos generalized this index by replacing 1 2 with any real number which is called the general randic index dvo ak et al introduced a modified version of randic index r g u v e g max d u d v 1 based on this recently knor et al introduced two generalizations r g u v e g min d u d v and r g u v e g max d u d v for any real number clearly the former is a lower bound for the general randic index and the latter is its upper bound knor et al studied extremal values of r g and r g and concluded some open problems in this paper we consider the open problems and give some comments and results some results for chemical trees are obtained

an f and m riesz theorem for planar vector fields we prove that solutions of the homogeneous equation lu 0 where l is a locally integrable vector field with smooth coefficients in two variables possess the f and m riesz property that is if omega is an open subset of the plane with smooth boundary u in c 1 omega satisfieslu 0 on omega has tempered growth at the boundary and its weak boundary value is a measure mu then mu is absolutely continuous with respect to lebesgue measure on the noncharacteristic portion of partial omega

on the distance based graph entropies in this note we prove a sharp upper bound for the distance based graph entropy introduced in chen et al 2014 furthermore we resolve one part of the conjecture proposed in the same paper and provide counterexamples for the other part

graph distance measures based on topological indices revisited graph distance measures based on topological indices have been already explored by dehmer et al also inequalities for those graph distance measures have been proved in this paper we continue studying such comparative graph measures based on the well known wiener index graph energy and randic index respectively we prove extremal properties of the graph distance measures for some special classes of graphs to demonstrate useful properties of the measures we also discuss numerical results to conclude the paper we state some open problems

the bipartite unicyclic graphs with the first n 3 4 largest matching energies the theory of matching energy of graphs since been proposed by gutman and wagner in 2012 has attracted more and more attention it is denoted by b n m the class of bipartite graphs with order n and size m in particular b n n denotes the set of bipartite unicyclic graphs which is an interesting class of graphs in this paper for odd n we characterize the bipartite unicyclic graphs with the first n 3 4 largest matching energies there is an interesting correspondence we conclude that the graph with the second maximal matching energy in b n n for odd n 11 is p n 6 which is the only graph attaining the maximum value of the energy among all the bipartite unicyclic graphs for n 16

wiener polarity index of dendrimers abstract network structures have many applications in biological information theory physical and social sciences research on network measurements is more and more important recently a number of topological indices have been studied as descriptors of complex networks the wiener polarity index has been introduced for a long time and known to be related to the cluster coefficient of networks in this paper we study the wiener polarity index of dendrimers which has many applications in other fields

structure based graph distance measures of high degree of precision in recent years evaluating graph distance has become more and more important in a variety of real applications and many graph distance measures have been proposed among all of those measures structure based graph distance measures have become the research focus due to their independence of the definition of cost functions however existing structure based graph distance measures have low degree of precision because only node and edge information of graphs are employed in these measures to improve the precision of graph distance measures we define substructure abundance vector sav to capture more substructure information of a graph furthermore based on sav we propose unified graph distance measures which are generalization of the existing structure based graph distance measures in general the unified graph distance measures can evaluate graph distance in much finer grain we also show that unified graph distance measures based on occurrence mapping and some of their variants are metrics finally we apply the unified graph distance metric and its variants to the population evolution analysis and construct distance graphs of marker networks in three populations which reflect the single nucleotide polymorphism snp linkage disequilibrium ld differences among these populations

digraphs with large maximum wiener index recently the concept of wiener index was extended to digraphs which are not necessarily strongly connected and it was shown that some fundamental results extend naturally within this concept this extension could be applicable in the topics of directed large networks particularly because with this measure one assigns finite values to the average distance and betweenness centrality of the nodes in a directed network it is not hard to show that among digraphs on n vertices the directed cycle c n achieves the maximum wiener index next we investigate digraphs with the second maximum wiener index one can consider this problem in the realm of all digraphs or restricted to those obtained by directing undirected graphs so called antisymmetric digraphs in both situations we obtain that such digraphs are constructed from c n n 6 by adding a single arc we conclude the paper with consideration for possible further works

the energy of a graph abstract the energy e g of a simple graph g is defined to be the sum of the absolute values of the eigen values of g if g is a k regular graph on n vertices then e g k k n 1 n k b 2 and this bound is sharp it is shown that for each 0 there exist infinitely many n for each of which there exists a k regular graph g of order n with k e g b 2 two graphs with the same number of vertices are equienergetic if they have the same energy we show that for any positive integer n 3 there exist two equienergetic graphs of order 4n that are not cospectral

a constructive proof of the fefferman stein decomposition of bmo rn let s vec r left left f r 1 f cdots r n f right f in h 1 left text r n text r right right subset h 1 left text r n text r n 1 right 22 1 begin gathered s vec r left vec g left g j right j 0 cdots n right in text bmo left text r n text r n 1 right sum limits j 0 n vec r j g j 0 text in bmo left text r n text r n 1 right left text r right end gathered 22 2 j 0

a history of graph entropy measures this survey seeks to describe methods for measuring the entropy of graphs and to demonstrate the wide applicability of entropy measures setting the scene with a review of classical measures for determining the structural information content of graphs we discuss graph entropy measures which play an important role in a variety of problem areas including biology chemistry and sociology in addition we examine relationships between selected entropy measures illustrating differences quantitatively with concrete examples

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

on the connective eccentricity index of trees and unicyclic graphs with given diameter abstract the connective eccentricity index cei of a graph g is defined as ce g v v g d v e v where e v d v denote the eccentricity and the degree of v respectively in this paper we investigate the connective eccentricity index of trees and unicyclic graphs on n vertices with diameter d we also characterize the extremal graphs

hosoya entropy of fullerene graphs abstract entropy based methods are useful tools for investigating various problems in mathematical chemistry computational physics and pattern recognition in this paper we introduce a general framework for applying shannon entropy to fullerene graphs and used it to investigate their properties we show that important physical properties of these molecules can be determined by applying hosoya entropy to their corresponding graphs

a measure of similarity between graph vertices applications to synonym extraction and web searching we introduce a concept of similarity between vertices of directed graphs let ga and gb be two directed graphs with respectively na and nb vertices we define an nb times na similarity matrix s whose real entry sij expresses how similar vertex j in ga is to vertex i in gb we say that sij is their similarity score the similarity matrix can be obtained as the limit of the normalized even iterates of sk 1 bskat btska where a and b are adjacency matrices of the graphs and s0 is a matrix whose entries are all equal to 1 in the special case where ga gb g the matrix s is square and the score sij is the similarity score between the vertices i and j of g we point out that kleinberg s hub and authority method to identify web pages relevant to a given query can be viewed as a special case of our definition in the case where one of the graphs has two vertices and a unique directed edge between them in analogy to kleinberg we show that our similarity scores are given by the components of a dominant eigenvector of a nonnegative matrix potential applications of our similarity concept are numerous we illustrate an application for the automatic extraction of synonyms in a monolingual dictionary

new highlights and a new centrality measure based on the adapted pagerank algorithm for urban networks a centrality measure with a practical application in urban network has been proposed the measure establishes a ranking of nodes focused on the topological data distribution the amount of data associated with the nodes of the network has a great importance the topology of the network and the information associated to the nodes are involved the adapted pagerank algorithm apa proposed by agryzkov et al provides us a method to establish a ranking of nodes in an urban network we can say that it constitutes a centrality measure in urban networks with the main characteristic that is able to consider the importance of data obtained from the urban networks in the process of computing the centrality of every node starting from the basic idea of this model we modify the construction of the matrix used for the classification of the nodes in order of importance in the apa model the data matrix is constructed from the original idea of pagerank vector given an equal chance to jump from one node to another regardless of the topological distance between nodes in the new model this idea is questioned a new matrix with the data network is constructed so that now the data from neighboring nodes are considered more likely than data from the nodes that are farther away in addition this new algorithm has the characteristic that depends on a parameter which allows us to decide the importance attached in the computation of the centrality to the topology of the network and the amount of data associated with the node various numerical experiments with a network of very small size are performed to test the influence of the data associated with the nodes depending always on the choice of the parameter also we check the differences between the values produced by the original apa model and the new one finally these measures are applied to a real urban network in which we perform a visual comparison of the results produced by the various measures calculated from the algorithms studied

inequalities for entropy based measures of network information content this paper presents a method for establishing relations between entropy based measures applied to graphs a special class of relations called implicit information inequalities or implicit entropy bounds is developed a number of entropy based measures of the structural information content of a graph have been developed over the past several decades but little attention has been paid to relations among these measures the research reported here aims to remedy this deficiency

on the general trigonometric sums weighted by character sums the main purpose of this paper is using the estimates for character sums and the mean value properties of dirichlet l function to study the hybrid mean value properties of the general trigonometric sums which is weighted by the 2kth and first power mean of character sums in 1 p 4 the two sharp asymptotic formulae are derived and some interesting connections between these sums are established

network entropies based on independent sets and matchings entropy based measures have been used to characterize the structure of complex networks various graph entropies based on different graph parameters have been proposed and studied so it has been intricate to decide which measure is the right one for solving a particular problem in this paper we introduce graph entropy measures based on independent sets and matchings of graphs the values of entropies of some special graphs are calculated and we draw several conclusions regrading the usability of the measures based on our findings we conclude that it should be demanding studying extremal values of the novel entropies

graph entropy based on the number of spanning forests of c cyclic graphs abstract graph entropies have been introduced to quantitatively measure the structural information content of graphs and networks they have plenty of applications in various fields utilizing the number of subgraphs to establish measures for determining the complexity of molecular graphs are also prevalent in the study of mathematical chemistry in this paper we develop a new graph entropy measure that is based on the number of spanning forests we prove explicit expressions for the entropy for trees unicyclic and bicyclic graphs and show that the cycle graph cn attains the maximal value of the entropy for unicyclic graphs with order n and large cycle lengths based on generating numerical results we conjecture extremal unicyclic graphs with respect to the entropy as well as we compare the values of our entropy for c cyclic graphs and generate graphs of bicyclic graphs and tricyclic graphs with 6 vertices for performing further research

on steiner degree distance of trees let g be a connected graph and u v w its vertices by du is denoted the degree of the vertex u by d u v the ordinary distance of the vertices u and v and by d u v w the steiner distance of u v w the degree distance dd of g is defined as the sum of terms d u d v d u v over all pairs of vertices of g as early as in the 1990s a linear relation was discovered between dd of trees and the wiener index we now consider sdd the steiner distance generalization of dd defined as the sum of terms d u d v d w d u v w over all triples of vertices of g also in this case a linear relation between sdd and the wiener index could be established

comparing large scale graphs based on quantum probability theory abstract in this paper a new measurement to compare two large scale graphs based on the theory of quantum probability is proposed an explicit form for the spectral distribution of the corresponding adjacency matrix of a graph is established our proposed distance between two graphs is defined as the distance between the corresponding moment matrices of their spectral distributions it is shown that the spectral distributions of their adjacency matrices in a vector state includes information not only about their eigenvalues but also about the corresponding eigenvectors moreover we prove that the proposed distance is graph invariant and sub structure invariant examples with various graphs are given and distances between graphs with few vertices are checked computational results for real large scale graphs show that its accuracy is better than any existing methods and time cost is extensively cheap

information theoretic measures of uhg graphs with low computational complexity we introduce a novel graph class we call universal hierarchical graphs uhg whose topology can be found numerously in problems representing e g temporal spacial or general process structures of systems for this graph class we show that we can naturally assign two probability distributions for nodes and for edges which lead us directly to the definition of the entropy and joint entropy and hence mutual information establishing an information theory for this graph class furthermore we provide some results under which conditions these constraint probability distributions maximize the corresponding entropy also we demonstrate that these entropic measures can be computed efficiently which is a prerequisite for every large scale practical application and show some numerical examples

topological mappings between graphs trees and generalized trees we present novel topological mappings between graphs trees and generalized trees that means between structured objects with different properties the two major contributions of this paper are first to clarify the relation between graphs trees and generalized trees a graph class recently introduced second these transformations provide a unique opportunity to transform structured objects into a representation that might be beneficial for a processing e g by machine learning techniques for graph classification

a similarity measure for graphs with low computational complexity we present and analyze an algorithm to measure the structural similarity of generalized trees a new graph class which includes rooted trees for this we represent structural properties of graphs as strings and define the similarity of two graphs as optimal alignments of the corresponding property stings we prove that the obtained graph similarity measures are so called backward similarity measures from this we find that the time complexity of our algorithm is polynomial and hence significantly better than the time complexity of classical graph similarity methods based on isomorphic relations

on a problem analogues to lehmer s and hurwitz zeta function the main purpose of this paper is to introduce a problem analogous to lehmer s and study a mean value of its error part with the weight of hurwitz zeta function by using the properties of character sums and the analytic method a sharp asymptotic formula was given and a connection between the problem analogous to lehmer s and hurwitz zeta function was established

boundary behavior of generalized analytic functions abstract we study the boundary properties of the solutions of the elliptic equation a z z u b z z u a z u b z u 0 under the assumption that a and b are holder continuous and a and b are in l p for some p 2 these properties include the h p property the f and m riesz property and the rudin carleson property

what is a complex graph abstract many papers published in recent years show that real world graphs g n m n nodes m edges are more or less complex in the sense that different topological features deviate from random graphs here we narrow the definition of graph complexity and argue that a complex graph contains many different subgraphs we present different measures that quantify this complexity for instance c 1 e the relative number of non isomorphic one edge deleted subgraphs i e deck size however because these different subgraph measures are computationally demanding we also study simpler complexity measures focussing on slightly different aspects of graph complexity we consider heuristically defined product measures the products of two quantities which are zero in the extreme cases of a path and clique and entropy measures quantifying the diversity of different topological features the previously defined network graph complexity measures medium articulation and offdiagonal complexity odc belong to these two classes we study odc measures in some detail and compare it with our new measures for all measures the most complex graph g c max has a medium number of edges between the edge numbers of the minimum and the maximum connected graph n 1 m g c max n n 1 2 interestingly for some measures c this number scales exactly with the geometric mean of the extremes m g c max n 2 n 1 n 1 5 all graph complexity measures are characterized with the help of different example graphs for all measures the corresponding time complexity is given finally we discuss the complexity of 33 real world graphs of different biological social and economic systems with the six computationally most simple measures including odc the complexities of the real graphs are compared with average complexities of two different random graph versions complete random graphs just fixed n m and rewired graphs with fixed node degrees