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There has been an upsurge of research on complex networks in recent years. The purpose of this paper is to study the mathematical properties of the random pentagonal chain networks PE C n with the help of graph theory. Based on the networks PE C n , we first obtain the expected value expressions of the Gutman index, Schultz index, multiplicative degree-Kirchhoff index, and additive degree-Kirchhoff index, and then, we get the explicit expression formulas of their variances. Finally, we find that their limiting distributions all have the probabilistic and statistical significance of normal distribution.

Let H n be the linear heptagonal networks with 2 n heptagons. We study the structure properties and the eigenvalues of the linear heptagonal networks. According to the Laplacian polynomial of H n , we utilize the method of decompositions. Thus, the Laplacian spectrum of H n is created by eigenvalues of a pair of matrices: L A and L S of order numbers 5 n + 1 and 4 n + 1 n ! / r ! n − r ! , respectively. On the basis of the roots and coefficients of their characteristic polynomials of L A and L S , we get not only the explicit forms of Kirchhoff index but also the corresponding total number of spanning trees of H n .

Resolving parameters is a fundamental area of combinatorics with applications not only to many branches of combinatorics but also to other sciences. In this article, we construct a class of Toeplitz graphs, and will be denoted by $T_{2n}(W)$, so that they are Cayley graphs. First, we review some of the features of this class of graphs. In fact, this class of graphs are vertex transitive, and by calculating the spectrum of the adjacency matrix related with them, we show that this class of graphs cannot be edge transitive. Moreover, we show that this class of graphs cannot be distance regular, and since the computing resolving parameters of a class of graphs such that are not distance regular is more difficult, then we regard this as justification for our focus on some resolving parameters. In particular, we determine the minimal resolving set, doubly resolving set and strong metric dimension for this class of graphs., Comment: arXiv admin note: substantial text overlap with arXiv:1905.10527

In the supply chain environment, agile distribution refined the division of operations, responded quickly to the customer needs of the mobile terminal, and promoted the construction of an intelligent logistics distribution system. In order to integrate the distribution resources of the distribution system of the logistics industry and maximize the overall benefits of distribution, this paper designs an intelligent logistics agile distribution model and optimizes the distribution strategy. Firstly, this paper puts forward a rationalization evaluation system based on the established logistics agile distribution model. Make full use of the fuzzy subsets of each level to quantify the fuzzy indexes to evaluate, and then comprehensively clarify the index attributes through the fuzzy transformation criteria to obtain the evaluation results. Secondly, aiming at the networked agile distribution organization form, the scientific location mathematical model of agile distribution center and the path mathematical model of the agile distribution are established. Finally, the uncertain influence of many objective conditions on distribution agility has been analyzed, and the dynamic optimization model of an agile distribution path with the concept of time window under random demand is constructed. Through the experimental analysis, the rationalization evaluation strategy and index of verification test results show that the model has significant optimization advantages, which not only improve the rationality and economic benefits of distribution, but also meet the time requirements of all users.

By taking the 16 cities in Anhui Province for evaluation, the main influencing factors and indicator system for integrated urban-rural development in the new era were explored, to build the BCC model, cross-efficiency model, and game cross-efficiency model of DEA. The above models were applied for empirical analysis and comparative study on the rural revitalization and urban-rural integration efficiency in Anhui Province, to summarize the conclusions efficiency and give suggestions based on the above calculations.

Graph theory is widely used in power network analysis, complex network, and engineering calculation. Stanley depth is a geometric invariant of the module which is closely related to an algebraic invariant called depth of the module. At first, we propose a generalization of classical gear graph and extended m − level gear graph and then establish general closed formulas for the sharp bounds of Stanley depth of quotient of edge ideals associated to extended m -level gear graph. We establish general closed formulas for the sharp bounds of Stanley depth of quotient of edge ideals associated to extended gear graph. We recover these bounds for Stanley depth of the quotient of edge ideals associated to classical gear graphs.

Chemical graph theory is the combination of mathematical graph theory and chemistry. To analyze the biocompatibility of the compounds, topological indices are used in the research of QSAR/QSPR studies. The degree-based entropy is inspired by Shannon’s entropy. The connectivity pattern such as planar octahedron network is used to predict physiochemical activity. In this article, we present some degree-based entropies of planar octahedron network.

Simulation and prediction of the scale change of fungal community. First, using the experimental data of a variety of fungal decomposition activities, a mathematical model of the decomposition rate and the relationship between the bacterial species was established, thereby revealing the internal mechanism of fungal decomposition activity in a complex environment. Second, based on the linear regression method and the principle of biodiversity, a model of fungal decomposition rate was constructed, and it was concluded that the interaction between mycelial elongation and moisture resistance could increase the fungal decomposition rate. Third, the differential equations are used to quantify the competitive relationship between different bacterial species, divide the boundaries of superior and inferior species, and simulate the long-term and short-term evolution trends of the community under the same initial environment. And an empirical analysis is made by taking the sudden change of the atmosphere affecting the evolution of the colony as an example. Finally, starting from summer, combining soil temperature, humidity, and fungal species data in five different environments such as arid and semiarid, a three-dimensional model and RBF neural network are introduced to predict community evolution. The study concluded that under given conditions, different strains are in short-term competition, and in the long-term, mutually beneficial symbiosis. Biodiversity is important for the biological regulation of nature.

The use of numerical numbers to represent molecular networks plays a crucial role in the study of physicochemical and structural properties of the chemical compounds. For some integer k and a network G , the networks S k G and R k G are its derived networks called as generalized subdivided and generalized semitotal point networks, where S k and R k are generalized subdivision and generalized semitotal point operations, respectively. Moreover, for two connected networks, G 1 and G 2 , G 1 G 2 S k and G 1 G 2 R k are T -sum networks which are obtained by the lexicographic product of T G 1 and G 2 , respectively, where T ε S k , R k . In this paper, for the integral value k ≥ 1 , we find exact values of the first and second Zagreb indices for generalized T -sum networks. Furthermore, the obtained findings are general extensions of some known results for only k = 1 . At the end, a comparison among the different generalized T -sum networks with respect to first and second Zagreb indices is also included.

Let Γ = V , E be a graph. If all the eigenvalues of the adjacency matrix of the graph Γ are integers, then we say that Γ is an integral graph. A graph Γ is determined by its spectrum if every graph cospectral to it is in fact isomorphic to it. In this paper, we investigate some algebraic properties of the Cayley graph Γ = Cay ℤ n , S , where n = p m ( p is a prime integer and m ∈ ℕ ) and S = a ∈ ℤ n | a , n = 1 . First, we show that Γ is an integral graph. Also, we determine the automorphism group of Γ . Moreover, we show that Γ and K v ▽ Γ are determined by their spectrum.

A molecular descriptor is a mathematical measure that associates a molecular graph with some real numbers and predicts the various biological, chemical, and structural properties of the underlying molecular graph. Wiener (1947) and Trinjastic and Gutman (1972) used molecular descriptors to find the boiling point of paraffin and total π -electron energy of the molecules, respectively. For molecular graphs, the general sum-connectivity and general Randić are well-studied fundamental topological indices (TIs) which are considered as degree-based molecular descriptors. In this paper, we obtain the bounds of the aforesaid TIs for the generalized F -sum graphs. The foresaid TIs are also obtained for some particular classes of the generalized F -sum graphs as the consequences of the obtained results. At the end, 3 D -graphical presentations are also included to illustrate the results for better understanding.

Financial big data are obtained by web crawler, and investors’ recognition abilities for risk and profit in online loan markets are researched using heteroskedastic Probit models. The conclusions are obtained as follows: First, the preference for the item is reflected directly in the time and indirectly in the number of participants for being full, and the larger the preference, the shorter the time and the fewer the participants. Second, investors can discriminate the default risk not reflected by the interest rate, and the bigger the default risk, the longer the time and the more participants being full. Third, investors can discriminate the pure return rate deducted from the maturity term and credit risk, and the higher the return, the shorter the time and the fewer the participants being full. Fourth, default risks are reflected well by online loan platform interest rates, and inventors do not choose the item blindly according to the interest rate but consider comprehensively the profit and the risk. In the future, interest rate liberalization should be deepened, the choosing function of interest rates should be played better, and the information disclosure, investor education, and investor effective usage of other information should be strengthened.

Topological index (TI) is a function that assigns a numeric value to a (molecular) graph that predicts its various physical and structural properties. In this paper, we study the sum graphs (S-sum, R-sum, Q-sum and T-sum) using the subdivision related operations and strong product of graphs which create hexagonal chains isomorphic to many chemical compounds. Mainly, the exact values of first general Zagreb index (FGZI) for four sum graphs are obtained. At the end, FGZI of the two particular families of sum graphs are also computed as applications of the main results. Moreover, the dominating role of the FGZI among these sum graphs is also shown using the numerical values and their graphical presentations.

Distance-based numeric parameters play a pivotal role in studying the structural aspects of networks which include connectivity, accessibility, centrality, clustering modularity, complexity, vulnerability, and robustness. Several tools like these also help to resolve the issues faced by the different branches of computer science and chemistry, namely, navigation, image processing, biometry, drug discovery, and similarities in chemical compounds. For this purpose, in this article, we are considering a family of networks that exhibits rotationally symmetric behaviour known as circular ladders consisting of triangular, quadrangular, and pentagonal faced ladders. We evaluate their upper bounds of fractional metric dimensions of the aforementioned networks.

Structure-based topological descriptors of chemical networks enable us the prediction of physico-chemical properties and the bioactivities of compounds through QSAR/QSPR methods. Topological indices are the numerical values to represent a graph which characterises the graph. One of the latest distance-based topological index is the Mostar index. In this paper, we study the Mostar index, Szeged index, PI index, ABC GG index, and NGG index, for chain oxide network COX n , chain silicate network CS n , ortho chain S n , and para chain Q n , for the first time. Moreover, analytically closed formulae for these structures are determined.

In this paper, we give the relation between the spectrum of strongly regular graph and its clique-inserted graph. The Laplacian spectrum and the signless Laplacian spectrum of clique-inserted graph of strongly regular graph are calculated. We also give formulae expressing the energy, Kirchoff index, and the number of spanning trees of clique-inserted graph of a strongly regular graph. And, clique-inserted graph of the triangular graph T t , which is a strongly regular graph, is enumerated.

Let G = V E , E G be a connected graph with vertex set V G and edge set E G . For a graph G, the graphs S(G), R(G), Q(G), and T(G) are obtained by applying the four subdivisions related operations S, R, Q, and T, respectively. Further, for two connected graphs G 1 and G 2 , G 1 + F G 2 are F -sum graphs which are constructed with the help of Cartesian product of F G 1 and G 2 , where F ∈ S , R , Q , T . In this paper, we compute the lower and upper bounds for the first Zagreb coindex of these F -sum (S-sum, R-sum, Q-sum, and T-sum) graphs in the form of the first Zagreb indices and coincides of their basic graphs. At the end, we use linear regression modeling to find the best correlation among the obtained results for the thirteen physicochemical properties of the molecular structures such as boiling point, density, heat capacity at constant pressure, entropy, heat capacity at constant time, enthalpy of vaporization, acentric factor, standard enthalpy of vaporization, enthalpy of formation, octanol-water partition coefficient, standard enthalpy of formation, total surface area, and molar volume.

In this article, an algorithm has been established to approximate parametric-parametric, explicit-implicit, and explicit-explicit surface intersection. Foremost, it extracts the characteristic points (boundary and turning points) from the sequence of intersection points and fits an optimal cubic spline curve to these points. Moreover, this paper utilizes genetic algorithm (GA) for optimization of shape parameters in the portrayal of cubic spline so that the error is minimal. The proposed algorithm is demonstrated with different types of surfaces to analyze its robustness and proficiency. In the end, all illustrations show the effectiveness of the algorithm which makes it more influential to resolve all complexities arises during intersection with a minimal error.

In chemical graph theory, graph invariants are usually referred to as topological indices. For a graph G , its vertex-degree-based topological indices of the form BID G = ∑ u v ∈ E G β d u , d v are known as bond incident degree indices, where E G is the edge set of G , d w denotes degree of an arbitrary vertex w of G , and β is a real-valued-symmetric function. Those BID indices for which β can be rewritten as a function of d u + d v − 2 (that is degree of the edge u v ) are known as edge-degree-based BID indices. A connected graph G is said to be r -apex tree if r is the smallest nonnegative integer for which there is a subset R of V G such that R = r and G − R is a tree. In this paper, we address the problem of determining graphs attaining the maximum or minimum value of an arbitrary BID index from the class of all r -apex trees of order n , where r and n are fixed integers satisfying the inequalities n − r ≥ 2 and r ≥ 1 .

In this study, we first introduce polygonal cylinder and torus using Cartesian products and topologically identifications and then find their Wiener and hyper-Wiener indices using a quick, interesting technique of counting. Our suggested mathematical structures could be of potential interests in representation of computer networks and enhancing lattice hardware security.

The normalized Laplacian plays an indispensable role in exploring the structural properties of irregular graphs. Let L n 8,4 represent a linear octagonal-quadrilateral network. Then, by identifying the opposite lateral edges of L n 8,4 , we get the corresponding Möbius graph M Q n 8,4 . In this paper, starting from the decomposition theorem of polynomials, we infer that the normalized Laplacian spectrum of M Q n 8,4 can be determined by the eigenvalues of two symmetric quasi-triangular matrices ℒ A and ℒ S of order 4 n . Next, owing to the relationship between the two matrix roots and the coefficients mentioned above, we derive the explicit expressions of the degree-Kirchhoff indices and the complexity of M Q n 8,4 .

It is one of the core problems in the study of chemical graph theory to study the topological index of molecular graph and the internal relationship between its structural properties and some invariants. In recent years, topological index has been gradually applied to the models of QSAR and QSPR . In this work, using the definition of the ABC index, AZI index, GA index, the multiplicative version of ordinary first Zagreb index, the second multiplicative Zagreb index, and Zagreb index, we calculate the degree-based topological indices of some networks. Then, the above indices’ formulas are obtained.

For the study and valuation of social graphs, which affect an extensive range of applications such as community decision-making support and recommender systems, it is highly recommended to sustain the resistance of a social graph G to active attacks. In this regard, a novel privacy measure, called the k,l-anonymity, is used since the last few years on the base of k-metric antidimension of G in which l is the maximum number of attacker nodes defining the k-metric antidimension of G for the smallest positive integer k. The k-metric antidimension of G is the smallest number of attacker nodes less than or equal to l such that other k nodes in G cannot be uniquely identified by the attacker nodes. In this paper, we consider four families of wheel-related social graphs, namely, Jahangir graphs, helm graphs, flower graphs, and sunflower graphs. By determining their k-metric antidimension, we prove that each social graph of these families is the maximum degree metric antidimensional, where the degree of a vertex is the number of vertices linked with that vertex.

A connected graph is called Hamilton-connected if there exists a Hamiltonian path between any pair of its vertices. Determining whether a graph is Hamilton-connected is an NP-complete problem. Hamiltonian and Hamilton-connected graphs have diverse applications in computer science and electrical engineering. The detour index of a graph is defined to be the sum of lengths of detours between all the unordered pairs of vertices. The detour index has diverse applications in chemistry. Computing the detour index for a graph is also an NP-complete problem. In this paper, we study the Hamilton-connectivity of convex polytopes. We construct three infinite families of convex polytopes and show that they are Hamilton-connected. An infinite family of non-Hamilton-connected convex polytopes is also constructed, which, in turn, shows that not all convex polytopes are Hamilton-connected. By using Hamilton connectivity of these families of graphs, we compute exact analytical formulas of their detour index.

A vertex w ∈ V H distinguishes (or resolves) two elements (edges or vertices) a , z ∈ V H ∪ E H if d w , a ≠ d w , z . A set W m of vertices in a nontrivial connected graph H is said to be a mixed resolving set for H if every two different elements (edges and vertices) of H are distinguished by at least one vertex of W m . The mixed resolving set with minimum cardinality in H is called the mixed metric dimension (vertex-edge resolvability) of H and denoted by m dim H . The aim of this research is to determine the mixed metric dimension of some wheel graph subdivisions. We specifically analyze and compare the mixed metric, edge metric, and metric dimensions of the graphs obtained after the wheel graphs’ spoke, cycle, and barycentric subdivisions. We also prove that the mixed resolving sets for some of these graphs are independent.

Tetrahedral network is considered as an effective tool to create the finite element network model of simulation, and many research studies have been investigated. The aim of this paper is to calculate several topological indices of the linear and circle tetrahedral networks. Firstly, the resistance distances of the linear tetrahedral network under different classifications have been calculated. Secondly, according to the above results, two kinds of degree-Kirchhoff indices of the linear tetrahedral network have been achieved. Finally, the exact expressions of Kemeny’s constant, Randic index, and Zagreb index of the linear tetrahedral network have been deduced. By using the same method, the topological indices of circle tetrahedral network have also been obtained.

A topological invariant is a numerical parameter associated with molecular graph and plays an imperative role in the study and analysis of quantitative structure activity/property relationships (QSAR/QSPR). The correlation between the entire π-electron energy and the structure of a molecular graph was explored and understood by the first Zagreb index. Recently, Liu et al. (2019) calculated the first general Zagreb index of the F-sum graphs. In the same paper, they also proposed the open problem to compute the general Randić index RαΓ=∑uv∈EΓdΓu×dΓvα of the F-sum graphs, where α∈R and dΓu denote the valency of the vertex u in the molecular graph Γ. Aim of this paper is to compute the lower and upper bounds of the general Randić index for the F-sum graphs when α∈N. We present numerous examples to support and check the reliability as well as validity of our bounds. Furthermore, the results acquired are the generalization of the results offered by Deng et al. (2016), who studied the general Randić index for exactly α=1.

In this article, we present several new inequalities involving the kappa-fractional integral for the integrable function F which satisfies one of the following conditions is a Lipschitz function. As applications, we establish new inequalities for the weighted arithmetic and generalized logarithmic means., Natural Science Foundation of China [11601140]; Scientific Research Fund of Hunan Provincial Education Department [16B047], This work was supported by the Natural Science Foundation of China (Grant no. 11601140) and the Scientific Research Fund of Hunan Provincial Education Department (Grant no. 16B047).

Reduced numbers play an important role in the study of modular group action on the PSL2,ℤ-subset of Qm/Q. For this purpose, we define new notions of equivalent, cyclically equivalent, and similar G-circuits in PSL2,ℤ-orbits of real quadratic fields. In particular, we classify PSL2,ℤ-orbits of Qm/Q=∪k∈NQ∗k2m containing G-circuits of length 6 and determine that the number of equivalence classes of G-circuits of length 6 is ten. We also employ the icosahedral group to explore cyclically equivalence classes of circuits and similar G-circuits of length 6 corresponding to each of these ten circuits. This study also helps us in classifying reduced numbers lying in the PSL2,ℤ-orbits.

Scientists are creating materials, for example, a carbon nanotube-based composite created by NASA that bends when a voltage is connected. Applications incorporate the use of an electrical voltage to change the shape (transform) of air ship wings and different structures. Topological indices are numbers related with molecular graphs to allow quantitative structure activity/property/poisonous relationships. Topological indices catch symmetry of molecular structures and give it a scientific dialect to foresee properties, for example, boiling points, viscosity, and the radius of gyrations. We compute M-polynomials of two nanotubes, SC5C7p,q and NPHXp,q. The closed form of M-polynomials for these nanotubes produces formulas of numerous degree-based topological indices which are functions relying on parameters of the structure and, in combination, decide properties of the concerned nanotubes. Moreover, we sketch our results by using Maple 2015 to see the dependence of our results on the involved parameters.

In mathematics, one always tries to get new structures from given ones. This also applies to the realm of graphs, where one can generate many new graphs from a given set of graphs. In this work, using knowledge of difference equations, we drive the explicit formulas for the number of spanning trees in the sequence of some graphs generated by a triangle by electrically equivalent transformations and rules of weighted generating function. Finally, we compare the entropy of our graphs with other studied graphs with average degree being 4, 5, and 6.

The degree-based topological indices are used to correlate the physical and chemical properties of a molecule with its chemical structure. Boron nanotubular structures are high-interest materials due to the presence of multicenter bonds and have novel electronic properties. These materials have some important issues in nanodevice applications like mechanical and thermal stability. Therefore, they require theoretical studies on the other properties. In this paper, we present certain degree-based topological indices such as ABC, the fourth ABC, GA, and the fifth GA indices for boron triangular and boron-α nanotubes.

We introduce an informative labeling algorithm for the vertices of a family of Koch networks. Each label consists of two parts: the precise position and the time adding to Koch networks. The shortest path routing between any two vertices is determined only on the basis of their labels, and the routing is calculated only by few computations. The rigorous solutions of betweenness centrality for every node and edge are also derived by the help of their labels. Furthermore, the community structure in Koch networks is studied by the current and voltage characterizations of its resistor networks.

We propose a new set-valued risk measure, which is called set-valued Haezendonck-Goovaerts risk measure. First, we construct the set-valued Haezendonck-Goovaerts risk measure and then provide an equivalent representation. The properties of the set-valued Haezendonck-Goovaerts risk measure are investigated, which show that the set-valued Haezendonck-Goovaerts risk measure is coherent. Finally, an example of set-valued Haezendonck-Goovaerts risk measure is given, which exhibits the fact that the set-valued average value at risk is a particular case of the set-valued Haezendonck-Goovaerts risk measures.