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

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

on tetravalent symmetric dihedrants let be a tetravalent x arc transitive cayley graph of dihedral group for x aut let xv be the stabilizer of x on vv has been determined when it is 2 arc transitive or one regular this paper studies the case where is one transitive but not one regular and gives it an exactly characterization as an application of this result we give a compete classification of such graphs when xv 24 by production a compete classification is given for the stabilizers of tetravalent symmetric cayley graphs whenever its order is less than 25

on the sharp bounds of bicyclic graphs regarding edge szeged index abstract for a given graph g its edge szeged index is denoted by s z e g e u v e g m u e m v e where mu e and mv e are the number of edges in g with distance to u less than to v and the number of edges in g further distance to u than v respectively in the paper the bounds of edge szeged index on bicyclic graphs are determined furthermore the graphs that achieve the bounds are completely characterized

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

a note on the first reformulated zagreb index let g v e v 1 2 n e e 1 e 2 e m be a simple graph with n vertices and m edges denote by d ei i 1 2 m an edge degree and by e m 1 i 1 m d e i 2 first reformulated zagreb index of graph g upper and lower bounds of graph invariant em1 are obtained

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

on edge rupture degree of graphs the edge rupture degree of an incomplete connected graph g is defined as r g m a x g s s m g s s e g g s 1 where g s and m g s respectively denote the number of components and the order of a largest component in g s this is a reasonable parameter to measure the vulnerability of networks as it takes into account both the amount of work done to damage the network and how badly the network is damaged in this paper firstly the relationships between the edge rupture degree and some other graph parameters namely the edge connectivity edge integrity edge toughness edge tenacity diameter the algebraic connectivity and the minimum degree are established after that the edge rupture degree of the middle graphs of path and cycle are given then we introduced the concept of r maximal graph and give some basic results of such graphs finally we introduce the concept of edge ruptured and strictly edge ruptured graph and we establish necessary and sufficient conditions for a graph to be edge ruptured and strictly edge ruptured respectively

characteristics polynomial of normalized laplacian for trees here we find the characteristics polynomial of normalized laplacian of a tree the coefficients of this polynomial are expressed by the higher order general randic indices for matching whose values depend on the structure of the tree we also find the expression of these indices for starlike tree and a double starlike tree hm p q moreover we show that two cospectral hm p q of the same diameter are isomorphic

complete characterization of bicyclic graphs with the maximum and second maximum degree kirchhoff index abstract the degree kirchhoff index or multiplicative degree kirchhoff index of a connected simple graph g is defined as s g u v v g d g u d g v r g u v where dg u is the degree of a vertex u in g and rg u v is the resistance distance between the vertices u and v in this paper we completely characterize the bicyclic graphs of order n 6 having the maximum degree kirchhoff index moreover the bicyclic graphs of order n 7 with the second maximum degree kirchhoff index have also been determined

on topological properties of the line graphs of subdivision graphs of certain nanostructures in the study of qsar qspr topological indices such as shultz index generalized randic index zagreb index general sum connectivity index atom bond connectivity abc index and geometric arithmetic ga index are exploited to estimate the bioactivity of chemical compounds a topological index attaches a chemical structure with a numeric number there are numerous applications of graph theory in this field of research in this paper we computed generalized randic general zagreb general sum connectivity abc ga abc4 and ga5 indices of the line graphs of 2d lattice nanotube and nanotorus of tuc4c8p q by using the concept of subdivision

nonlocality distillation and postquantum theories with trivial communication complexity we first present a protocol for deterministically distilling nonlocality building upon a recent result of forster et al phys rev lett 102 120401 2009 our protocol which is optimal for two copy distillation works efficiently for a specific class of postquantum nonlocal boxes which we term correlated nonlocal boxes in the asymptotic limit all correlated nonlocal boxes are distilled to the maximally nonlocal box of popescu and rohrlich then taking advantage of a result of brassard et al phys rev lett 96 250401 2006 we show that all correlated nonlocal boxes make communication complexity trivial and therefore appear very unlikely to exist in nature astonishingly some of these nonlocal boxes are arbitrarily close to the set of classical correlations this result therefore gives new insight to the problem of why quantum nonlocality is limited

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 extremal values of some topological indices in bipartite graphs with a given matching number let i g be a topological index of a graph if i g e i g or i g e i g respectively for each edge e g then i g decreases or increases respectively with addition of edges in this paper we determine the extremal values of some topological indices which decrease or increase with addition of edges and characterize the corresponding extremal graphs in bipartite graphs with a given matching number

vertex degree based topological indices of hexagonal systems with equal number of edges in this paper we find extremal values of vertex degree based topological indices over m the set of hexagonal systems with m edges the main idea consists in constructing hexagonal systems with maximal number of inlets in m which have simultaneously minimal number of hexagons also we show that the convex spiral sh has extremal ti value over 3h 12h3

graphs with fixed number of pendent vertices and minimal zeroth order general randic index we investigate the graph with minimal zeroth order general randic index in terms of its order n pendent number n1 and cyclomatic number 0 extremal graphs were completely characterized for cases of 0 1 2 which can be directly extended for graphs with cyclomatic number 3

nordhaus gaddum type results for graph irregularities abstract a graph whose vertices have the same degree is called regular otherwise the graph is irregular in fact various measures of irregularity have been proposed and examined for a given graph g v e with v v 1 v 2 v n and edge set e g di is the vertex degree where 1 i n the irregularity of g is defined by i r r g v i v j e g d i d j a similar measure can be defined by i r r 2 g v i v j e g d i d j 2 the total irregularity of g is defined by i r r t g 1 2 v i v j v g d i d j the variance of the vertex degrees is defined v a r g 1 n i 1 n d i 2 2 m n 2 in this paper we present some nordhaus gaddum type results for these measures and draw conclusions

upper bound on the sum of powers of the degrees of graphs with few crossings per edge abstract a graph is q planar if it can be drawn in the plane so that each edge is crossed by at most q other edges for fixed integers q 1 and k 2 it is proven that 2 n 1 k o n is an asymptotically tight upper bound on the sum of the k th powers of the degrees of any simple q planar graph with order n as a result an open problem listed at the end of the paper j czap j harant d hudak discrete appl math 165 2014 146 151 is solved

on certain topological indices of the line graph of subdivision graphs in qsar qspr study topological indices such as shultz index generalized randic index zagreb index general sum connectivity index atom bond connectivity abc index and geometric arithmetic ga index are utilized to guess the bioactivity of chemical compounds a topological index in fact relates a chemical structure with a numeric number graph theory has established a significant use in this area of research in this paper we computed abc4 and ga5 indices of the line graph of tadpole wheel and ladder graphs using the notion of subdivision

the general randic index of trees with given number of pendent vertices the general randic index of a graph g is defined as r g u v e g d u d v where d u denotes the degree of a vertex u in g and is a real number in this paper we determine the maximum general randic indices of trees and chemical trees with n vertices and k pendent vertices for 4 k n 2 3 and 0 0 where 0 0 5122 is the unique non zero root of the equation 6 4 20 9 10 12 16 5 24 0 the corresponding extremal graphs are also characterized

spectral properties of geometric arithmetic index the concept of geometric arithmetic index was introduced in the chemical graph theory recently but it has shown to be useful one of the main aims of algebraic graph theory is to determine how or whether properties of graphs are reflected in the algebraic properties of some matrices the aim of this paper is to study the geometric arithmetic index ga1 from an algebraic viewpoint since this index is related to the degree of the vertices of the graph our main tool will be an appropriate matrix that is a modification of the classical adjacency matrix involving the degrees of the vertices moreover using this matrix we define a ga laplacian matrix which determines the geometric arithmetic index of a graph and satisfies properties similar to the ones of the classical laplacian matrix

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

on zagreb indices zagreb polynomials of some nanostar dendrimers dendrimers are highly branched organic macromolecules with successive layers or generations of branch units surrounding a central core ashrafi and mirzargar 2008 3 chen et al 2014 7 klajnert and bryszewska 2001 26 yamamoto et al 2002 36 these are key molecules in nanotechnology and can be put to good use topological indices are numbers associated with molecular graphs for the purpose of allowing quantitative structure activity property toxicity relationships these topological indices correlate certain physico chemical properties like boiling point stability strain energy etc of chemical compounds in this paper we determine hyper zagreb index first multiple zagreb index second multiple zagreb index and zagreb polynomials for some nanostar dendrimers

on the graovac ghorbani index of graphs the graovac ghorbani index abcgg was introduced by graovac and ghorbani 2010 in this paper we present some upper and lower bounds on the abcgg index and characterize the extremal graphs

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

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

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

the blowup formula for donaldson invariants k k which is calculated in 4 below this formula has been the target of much recent work the abstract fact that there exists such a formula which is independent of x was first proved by c taubes using techniques of t j bryan b and p ozsvath o have independently calculated the coefficients throughb10 x quite recently j morgan and ozsvath have announced a scheme which can recursively compute all of the bk x the special case of the blowup formula for manifolds of simple type see 5 below was first given by p kronheimer and t mrowka however none of the techniques in these cases approach the simplicity of that offered here before presenting the formula we shall first establish notation for the donaldson in variants of a simply connected 4 manifold x with b 1 and odd the hypothesis of simple connectivity is not necessary but makes the exposition easier an orientation of x together with an orientation of h 2 x r is called a homology orientation of x such a homology orientation determines the su 2 donaldson invariant a linear function d dx a x sym h0 x h2 x r

multi user non locality amplification non local correlations are among the most fascinating features of quantum theory from the point of view of information such correlations although not allowing for signaling are unexplainable by pre shared information the correlations have applications in cryptography communication complexity and sit at the very heart of many attempts of understanding quantum theory and its limits in terms of classical information in this paper the question is crucial whether such correlations can be amplified or distilled i e whether and how weak correlations can be used for generating a smaller amount of stronger whereas the question has been studied quite extensively for bipartite correlations yielding both pessimistic and optimistic results only little is known in the multi partite case we introduce a general framework of reductions between multi party input output systems within this formalism we show that a natural n party generalization of the well known popescu rohrlich box can be distilled by an adaptive protocol to the algebraic maximum we use this result further to show that a much broader class of correlations including all purely three partite correlations can be distilled from arbitrarily weak to almost maximal strength with partial communication i e using only a subset of the channels required for the creation of the same correlation from scratch alternatively this means that arbitrarily weak non local correlations can have a communication value in the context of the generation of maximal non locality

on nonlocality as a resource theory and nonlocality measures with the advent of device independent quantum information processing nonlocality is nowadays regarded as a resource that can be used for various tasks using the analogy of entanglement theory we approach nonlocality from this perspective in order to do so we analyze in full detail the operations that can be implemented in this scenario and under which nonlocality cannot increase this provides a theoretical basis from which to study how to order and quantify nonlocal behavior finally we review several nonlocality measures and discuss their validity from this point of view this article is part of a special issue of journal of physics a mathematical and theoretical devoted to 50 years of bell s theorem

sqbc an efficient subgraph matching method over large and dense graphs recent progress in biology and computer science have generated many complicated networks most of which can be modeled as large and dense graphs developing effective and efficient subgraph match methods over these graphs is urgent meaningful and necessary although some excellent exploratory approaches have been proposed these years they show poor performances when the graphs are large and dense this paper presents a novel subgraph query technique based on clique feature called sqbc which integrates the carefully designed clique encoding with the existing vertex encoding 40 as the basic index unit to reduce the search space furthermore sqbc optimizes the subgraph isomorphism test based on clique features extensive experiments over biological networks rdf dataset and synthetic graphs have shown that sqbc outperforms the most popular competitors both in effectiveness and efficiency especially when the data graphs are large and dense

on the edge version atom bond connectivity and geometric arithmetic indices of certain graph operations let dl g e be the degree of an edge e in line graph l g of a graph g the edge versions of atom bond connectivity abce and geometric arithmetic gae indices of g are defined as efe l g dl g e dl g f 2dl g e dl g f and gae g efe l g 2dl g e dl g f dl g e dl g f in this paper we study the abce and gae indices for joint graphs and certain graph operations

integral trees with diameter four a graph is called integral if all eigenvalues of its adjacency matrix consist entirely of integers in this paper we investigate integral trees s r m i s a 1 a 2 a s m 1 m 2 m s of diameter 4 with s 3 4 5 6 such integral trees are found by using a computer search or solving the diophantine equations new sufficient conditions for a construction of infinite families of integral trees s r m i s b 1 b s m 1 ms of diameter 4 from given integral trees s r m i s a 1 a s m 1 ms of diameter 4 are given further using these conditions we construct infinitely many new classes of integral trees s r m i s b 1 b s m 1 ms of diameter 4 with s 3 4 5 6 finally we propose two basic open problems about integral trees of diameter 4 for further study

the non locality of n noisy popescu rohrlich boxes we quantify the amount of non locality contained in n noisy versions of the so called popescu rohrlich boxes prbs i e bipartite systems violating the chsh bell inequality maximally following the approach by elitzur popescu and rohrlich we measure the amount of non locality of a system by representing it as a convex combination of a local behaviour with maximal possible weight and a non signalling system we show that the local part of n systems each of which approximates a prb with probability 1 e is of order en 2 in the isotropic and equal to 3e n in the maximally biased case

on extremal zagreb indices of trees with given domination number first m1 and second m2 zagreb indices are graph invariants that originate from chemical researches on total electron energy of conjugated molecules there is a legion of articles dealing with these two indices this paper presents upper bounds on zagreb indices of trees in terms of domination number these are strict bounds and extremal trees are characterized in addition a lower bound for the first zagreb index of trees with a given domination number is determined and the extremal trees are characterized as well finally using previously known upper bound for harary index h in terms of m1 and m2 a unique tree with given domination number that maximizes h is characterized

instanton counting and donaldson invariants for a smooth projective toric surface we determine the donaldson invariants and their wallcrossing in terms of the nekrasov partition function using the solution of the nekrasov conjecture 33 38 3 and its refinement 34 we apply this result to give a generating function for the wallcrossing of donaldson invariants of good walls of simply connected projective surfaces with b 1 in terms of modular forms this formula was proved earlier in 19 more generally for simply connected 4 manifolds with b 1 assuming the kotschick morgan conjecture and it was also derived by physical arguments in 31

distilling non locality two parts of an entangled quantum state can have a correlation in their joint behavior under measurements that is unexplainable by shared classical information such correlations are called non local and have proven to be an interesting resource for information processing since non local correlations are more useful if they are stronger it is natural to ask whether weak non locality can be amplified we give an affirmative answer by presenting the first protocol for distilling non locality in the framework of generalized non signaling theories our protocol works for both quantum and non quantum correlations this shows that in many contexts the extent to which a single instance of a correlation can violate a chsh inequality is not a good measure for the usefulness of non locality a more meaningful measure follows from our results

a computational approach to construct a multivariate complete graph invariant in this paper we present a computational approach for finding complete graph invariants specifically we generate exhaustive sets of connected non isomorphic graphs with 9 and 10 vertices and demonstrate that a 97 dimensional multivariate graph invariant is capable to distinguish each of the non isomorphic graphs furthermore in order to tame the computational complexity of the problem caused by the vast number of graphs e g involving over 10 million networks with 10 vertices we suggest a low dimensional iterative procedure that is based on highly discriminative individual graph invariants we show that also this computational approach leads to a perfect discrimination overall our numerical results prove the existence of such graph invariants for networks with 9 and 10 vertices furthermore we show that our iterative approach has a polynomial time complexity

ordering chemical graphs by randic and sum connectivity numbers abstract let g be a graph with edge set e g the randic and sum connectivity indices of g are defined as r g u v e g 1 d e g g u d e g g v and s c i g u v e g 1 d e g g u d e g g v respectively where degg u denotes the vertex degree of u in g in this paper the extremal randic and sum connectivity index among all n vertex chemical trees n 13 connected chemical unicyclic graphs n 7 connected chemical bicyclic graphs n 6 and connected chemical tricyclic graphs n 8 were presented

further results regarding the sum of domination number and average eccentricity the average eccentricity of a graph g denoted by ecc g is the mean value of eccentricities of all vertices of g let dn i be the n vertex tree obtained from a path p n 1 v 1 v 2 v n 1 by attaching a pendent vertex to vi in 13 it was shown that the maximum value for the sum of domination number and average eccentricity among n vertex connected graphs is attained by dn 3 when n 0 mod 3 and attained by the path pn when n 0 mod 3 in this paper we will further determine the second maximum value for the sum of domination number and average eccentricity among n vertex connected graphs it is interesting that the graphs attaining that second maximum value have three cases which is dn 6 when n 0 mod 3 dn 3 when n 1 mod 3 and tn when n 2 mod 3 where tn is the n vertex tree obtained from a path p n 2 v 1 v 2 v n 2 by attaching a pendent vertex to v3 and a pendent vertex to v n 4

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

bond incident degree bid indices of polyomino chains this work is devoted to establish a general expression for calculating the bond incident degree bid indices of polyomino chains and to characterize the extremal polyomino chains with respect to several well known bid indices from the derived results all the results of the papers an and xiong 2016 deng et al 2014 yarahmadi et al 2012 and some results of the papers rada 2014 ali et al 2015 are obtained as corollaries

on the wiener polarity index of graphs the wiener polarity index wp g of a graph g is the number of unordered pairs of vertices u v in g such that the distance between u and v is equal to 3 very recently zhang and hu studied the wiener polarity index in y zhang y hu 2016 38 in this short paper we establish an upper bound on the wiener polarity index in terms of hosoya index and characterize the corresponding extremal graphs moreover we obtain nordhaus gaddum type results for wp g our lower bound on w p g w p g is always better than the previous lower bound given by zhang and hu

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

the weighted vertex pi index of n m graphs with given diameter abstract the weighted vertex pi index of a graph g is defined by p i w g e u v e g d g u d g v n u e g n v e g where nu e g denotes the number of vertices in g whose distance to the vertex u is smaller than the distance to the vertex v in this paper we give the upper bound and the corresponding extremal graphs on the weighted vertex pi index of n m graphs with diameter d the lower bound and the corresponding extremal graphs on the first zagreb index and the weighted vertex pi index of trees with diameter d are given by two procedures the extremal graphs given by the two procedures are also the extremal graphs which attain the lower bound on the first zagreb index among all connected graphs with n vertices and diameter d

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

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