Symmetry, Volume 10, Issue 8 (August 2018) – 56 articles

The generating functions of fourteen families of generalized Chebyshev polynomials related to rank two Lie algebras $A_2$ , $C_2$ and $G_2$ are explicitly developed. There exist two classes of the orthogonal polynomials corresponding to the symmetric and antisymmetric orbit functions of each rank two algebra. The Lie algebras $G_2$ and $C_2$ admit two additional polynomial collections arising from their hybrid character functions. The admissible shift of the weight lattice permits the construction of a further four shifted polynomial classes of $C_2$ and directly generalizes formation of the classical univariate Chebyshev polynomials of the third and fourth kinds. Explicit evaluating formulas for each polynomial family are derived and linked to the incomplete exponential Bell polynomials. Full article

BM123-64 block cipher, which was proposed by Minh, N.H. and Bac, D.T. in 2014, was designed for high speed communication applications factors. It was constructed in hybrid controlled substitution–permutation network (CSPN) models with two types of basic controlled elements (CE) in distinctive designs. This cipher is based on switchable data-dependent operations (SDDO) and covers dependent-operations suitable for efficient primitive approaches for cipher constructions that can generate key schedule in a simple way. The BM123-64 cipher has advantages including high applicability, flexibility, and portability with different algorithm selection for various application targets with internet of things (IoT) as well as secure protection against common types of attacks, for instance, differential attacks and linear attacks. However, in this paper, we propose methods to possibly exploit the BM123-64 structure using related-key attacks. We have constructed a high probability related-key differential characteristics (DCs) on a full eight rounds of BM123-64 cipher. The related-key amplified boomerang attack is then proposed on all three different cases of operation-specific designs with effective results in complexity of data and time consumptions. This study can be considered as the first cryptographic results on BM123-64 cipher. Full article

Key agreement between two constrained Internet of Things (IoT) devices that have not met each other is an essential feature to provide in order to establish trust among its users. Physical Unclonable Functions (PUFs) on a device represent a low cost primitive exploiting the unique random patterns in the device and have been already applied in a multitude of applications for secure key generation and key agreement in order to avoid an attacker to take over the identity of a tampered device, whose key material has been extracted. This paper shows that the key agreement scheme of a recently proposed PUF based protocol, presented by Chatterjee et al., for Internet of Things (IoT) is vulnerable for man-in-the-middle, impersonation, and replay attacks in the Yao–Dolev security model. We propose an alternative scheme, which is able to solve these issues and can provide in addition a more efficient key agreement and subsequently a communication phase between two IoT devices connected to the same authentication server. The scheme also offers identity based authentication and repudiation, when only using elliptic curve multiplications and additions, instead of the compute intensive pairing operations. Full article

In this paper, the geometric modeling and virtual reconstruction of the double-acting steam engine designed by Agustín de Betancourt in 1789 are shown. For this, the software Autodesk Inventor Professional is used, which has allowed us to obtain its geometric documentation. The material for the research is available on the website of the Betancourt Project of the Canary Orotava Foundation for the History of Science. Almost all parts of the steam engine are drawn on the sheets, but due to the absence of scale and space, it is insufficient to obtain an accurate and reliable 3D CAD (Computer-Aided Design) model. For this reason a graphic scale has been adopted so that the dimensions of the elements are coherent. Also, it has been necessary to make some dimensional and geometric hypotheses, as well as restrictions of movement (degrees of freedom). Geometric modeling has made it possible to know that the system is balanced with the geometric center of the rocker arm shaft, and presents an energetic symmetry whose axis is the support of the parallelogram where the shaft rests: calorific energy to the left and mechanical energy to the right, with the rocker arm acting as a transforming element from one to the other. Full article

The concept of a commutative generalized neutrosophic ideal in a $BCK$ -algebra is proposed, and related properties are proved. Characterizations of a commutative generalized neutrosophic ideal are considered. Also, some equivalence relations on the family of all commutative generalized neutrosophic ideals in $BCK$ -algebras are introduced, and some properties are investigated. Full article

Electric multiple unit (EMU) trains’ high-level maintenance planning is a discrete problem in mathematics. The high-level maintenance process of the EMU trains consumes plenty of time. When the process is undertaken during peak periods of the passenger flow, the transportation demand may not be fully satisfied due to the insufficient supply of trains. In contrast, if the process is undergone in advance, extra costs will be incurred. Based on the practical requirements of high-level maintenance, a 0–1 programming model is proposed. To simplify the description of the model, candidate sets of delivery dates, i.e., time windows, are generated according to the historical data and maintenance regulations. The constraints of the model include maintenance regulations, the passenger transportation demand, and capacities of workshop. The objective function is to minimize the mileage losses of all EMU trains. Moreover, a modified particle swarm algorithm is developed for solving the problem. Finally, a real-world case study of Shanghai Railway is conducted to demonstrate the proposed method. Computational results indicate that the (approximate) optimal solution can be obtained successfully by our method and the proposed method significantly reduces the solution time to 500 s. Full article

Higher order Schwarzian derivatives for normalized univalent functions were first considered by Schippers, and those of convex functions were considered by Dorff and Szynal. In the present investigation, higher order Schwarzian derivatives for the Janowski star-like and convex functions are considered, and sharp bounds for the first three consecutive derivatives are investigated. The results obtained in this paper generalize several existing results in this direction. Full article

The occurrence rates of melanoma are rising rapidly, which are resulting in higher death rates. However, if the melanoma is diagnosed in Phase I, the survival rates increase. The segmentation of the melanoma is one of the largest tasks to undertake and achieve when considering both beneath and over the segmentation. In this work, a new approach based on the artificial bee colony (ABC) algorithm is proposed for the detection of melanoma from digital images. This method is simple, fast, flexible, and requires fewer parameters compared with other algorithms. The proposed approach is applied on the PH2, ISBI 2016 challenge, the ISBI 2017 challenge, and Dermis datasets. These bases contained images are affected by different abnormalities. The formation of the databases consists of images collected from different sources; they are bases with different types of resolution, lighting, etc., so in the first step, the noise was removed from the images by using morphological filtering. In the next step, the ABC algorithm is used to find the optimum threshold value for the melanoma detection. The proposed approach achieved good results in the conditions of high specificity. The experimental results suggest that the proposed method accomplished higher performance compared to the ground truth images supported by a Dermatologist. For the melanoma detection, the method achieved an average accuracy and Jaccard’s coefficient in the range of 95.24–97.61%, and 83.56–85.25% in these four databases. To show the robustness of this work, the results were compared to existing methods in the literature for melanoma detection. High values for estimation performance confirmed that the proposed melanoma detection is better than other algorithms, which demonstrates the highly differential power of the newly introduced features. Full article

Fault diagnosis is an important issue in various fields and aims to detect and identify the faults of systems, products, and processes. The cause of a fault is complicated due to the uncertainty of the actual environment. Nevertheless, it is difficult to consider uncertain factors adequately with many traditional methods. In addition, the same fault may show multiple features and the same feature might be caused by different faults. In this paper, a neutrosophic set based fault diagnosis method based on multi-stage fault template data is proposed to solve this problem. For an unknown fault sample whose fault type is unknown and needs to be diagnosed, the neutrosophic set based on multi-stage fault template data is generated, and then the generated neutrosophic set is fused via the simplified neutrosophic weighted averaging (SNWA) operator. Afterwards, the fault diagnosis results can be determined by the application of defuzzification method for a defuzzying neutrosophic set. Most kinds of uncertain problems in the process of fault diagnosis, including uncertain information and inconsistent information, could be handled well with the integration of multi-stage fault template data and the neutrosophic set. Finally, the practicality and effectiveness of the proposed method are demonstrated via an illustrative example. Full article

The notions of neutrosophy, neutrosophic algebraic structures, neutrosophic duplet and neutrosophic triplet were introduced by Florentin Smarandache. In this paper, the neutrosophic duplets of $Z_{p^n}$ , $Z_{pq}$ and $Z_{p_1{p}_2\dots p_n}$ are studied. In the case of $Z_{p^n}$ and $Z_{pq}$ , the complete characterization of neutrosophic duplets are given. In the case of $Z_{p_1\dots p_n}$ , only the neutrosophic duplets associated with $p_i$ s are provided; $i=1,2,\dots ,n$ . Some open problems related to neutrosophic duplets are proposed. Full article

The human visual system uses priors to convert an ill-posed inverse problem of 3D shape recovery into a well-posed one. In previous studies, we have demonstrated the use of priors like symmetry, compactness and minimal surface in the perception of 3D symmetric shapes. We also showed that binocular perception of symmetric shapes can be well modeled by the above-mentioned priors and binocular depth order information. In this study, which used a shape-matching task, we show that these priors can also be used to model perception of near-symmetrical shapes. Our near-symmetrical shapes are asymmetrical shapes obtained from affine distortions of symmetrical shapes. We found that the perception of symmetrical shapes is closer to veridical than the perception of asymmetrical shapes. We introduce a metric to measure asymmetry of abstract polyhedral shapes, and a similar metric to measure shape dissimilarity between two polyhedral shapes. We report some key observations obtained by analyzing the data from the experiment. A website was developed with all the shapes used in the experiment, along with the shapes recovered by the subject and the shapes recovered by the model. This website provides a qualitative analysis of the effectiveness of the model and also helps demonstrate the goodness of the shape metric. Full article

Non-pairwise additive three-body dispersion potentials dependent upon one or more electric octupole moments are evaluated using the theory of molecular quantum electrodynamics. To simplify the perturbation theory calculations, an effective two-photon interaction Hamiltonian operator is employed. This leads to only third-order theory being required to evaluate energy shifts instead of the usual sixth-order formula, and the summation over six time-ordered sequences of virtual photon creation and annihilation events. Specific energy shifts computed include DD-DD-DO, DD-DO-DO, DO-DO-DO, and DD-DO-OO terms, where D and O are electric dipole and octupole moments, respectively. The formulae obtained are applicable to an arbitrary arrangement of the three particles, and we present explicit results for the equilateral triangle and collinear configurations, which complements the recently published DD-DD-OO potential. In this last case it was found that the contribution from the octupole weight-1 term could be viewed as a higher-order correction to the triple-dipole dispersion potential DD-DD-DD. In a similar fashion the octupole moment is decomposed into its irreducible components of weights-1 and -3, enabling insight to be gained into the potentials obtained in this study. Dispersion interaction energies proportional to mixed dipole-octupole polarisabilities, for example, are found to depend only on the weight-1 octupole moment for isotropic species and are retarded. Additional approximations are necessary in the evaluation of wave vector integrals for these cases in order to yield energy shifts that are valid in the near-zone. Full article

Unsupervised machine learning and knowledge discovery from large-scale datasets have recently attracted a lot of research interest. The present paper proposes a distributed big data clustering approach-based on adaptive density estimation. The proposed method is developed-based on Apache Spark framework and tested on some of the prevalent datasets. In the first step of this algorithm, the input data is divided into partitions using a Bayesian type of Locality Sensitive Hashing (LSH). Partitioning makes the processing fully parallel and much simpler by avoiding unneeded calculations. Each of the proposed algorithm steps is completely independent of the others and no serial bottleneck exists all over the clustering procedure. Locality preservation also filters out the outliers and enhances the robustness of the proposed approach. Density is defined on the basis of Ordered Weighted Averaging (OWA) distance which makes clusters more homogenous. According to the density of each node, the local density peaks will be detected adaptively. By merging the local peaks, final cluster centers will be obtained and other data points will be a member of the cluster with the nearest center. The proposed method has been implemented and compared with similar recently published researches. Cluster validity indexes achieved from the proposed method shows its superiorities in precision and noise robustness in comparison with recent researches. Comparison with similar approaches also shows superiorities of the proposed method in scalability, high performance, and low computation cost. The proposed method is a general clustering approach and it has been used in gene expression clustering as a sample of its application. Full article

In this work, we study the completely integrable sixth-order nonlinear Ramani equation. By applying the Lie symmetry analysis technique, the Lie point symmetries and the optimal system of one-dimensional sub-algebras of the equation are derived. The optimal system is further used to derive the symmetry reductions and exact solutions. In conjunction with the Riccati Bernoulli sub-ODE (RBSO), we construct the travelling wave solutions of the equation by solving the ordinary differential equations (ODEs) obtained from the symmetry reduction. We show that the equation is nonlinearly self-adjoint and construct the conservation laws (CL) associated with the Lie symmetries by invoking the conservation theorem due to Ibragimov. Some figures are shown to show the physical interpretations of the acquired results. Full article

Secret image sharing (SIS) with small-sized shadow images has many benefits, such as saving storage space, improving transmission time, and achieving information hiding. When adjacent pixel values in an image are similar to each other, the secret image will be leaked when all random factors of an SIS scheme are utilized for achieving small sizes of shadow images. Most of the studies in this area suffer from an inevitable problem: auxiliary encryption is crucial in ensuring the security of those schemes. In this paper, an SIS scheme with small-sized shadow images based on the Chinese remainder theorem (CRT) is proposed. The size of shadow images can be reduced to nearly $1/k$ of the original secret image. By adding random bits to binary representations of the random factors in the CRT, auxiliary encryption is not necessary for this scheme. Additionally, reasonable modifications of the random factors make it possible to incorporate all advantages of the CRT as well, including a ( $k,n$ ) threshold, lossless recovery, and low computation complexity. Analyses and experiments are provided to demonstrate the effectiveness of the proposed scheme. Full article

We consider the constrained ordered weighted averaging (OWA) aggregation problem with a single constraint and lower bounded variables. For the three-dimensional constrained OWA aggregation problem with lower bounded variables, we present four types of solution depending on the number of zero elements. According to the computerized experiment we perform, the lower bounds can affect the solution types, thereby affecting the optimal solution of the three-dimensional constrained OWA aggregation problem with lower bounded variables. Full article

At the dawn of big data and 5G networks, end-to-end communication with large amounts of data between mobile devices is difficult to be implemented through the traditional face-to-face transmission mechanism in social networks. Consequently, opportunistic social networks proposed that message applications should choose proper relay nodes to perform effective data transmission processes. At present, several routing algorithms, based on node similarity, attempt to use the contextual information related to nodes and the special relationships between them to select a suitable relay node among neighbors. However, when evaluating the similarity degree between a pair of nodes, most existing algorithms in opportunistic social networks pay attention to only a few similar factors, and even ignore the importance of mobile similarity in the data transmission process. To improve the transmission environment, this study establishes a fuzzy routing-forwarding algorithm (FCNS) exploiting comprehensive node similarity (the mobile and social similarities) in opportunistic social networks. In our proposed scheme, the transmission preference of the node is determined through the fuzzy evaluation of mobile and social similarities. The suitable message delivery decision is made by collecting and comparing the transmission preference of nodes, and the sustainable and stable data transmission process is performed through the feedback mechanism. Through simulations and the comparison of social network algorithms, the delivery ratio in the proposed algorithm is 0.85 on average, and the routing delay and network overhead of this algorithm are always the lowest. Full article

A novel global harmony search (NGHS) algorithm, as proposed in 2010, is an improved algorithm that combines the harmony search (HS), particle swarm optimization (PSO), and a genetic algorithm (GA). Moreover, the fixed parameter of mutation probability was used in the NGHS algorithm. However, appropriate parameters can enhance the searching ability of a metaheuristic algorithm, and their importance has been described in many studies. Inspired by the adjustment strategy of the improved harmony search (IHS) algorithm, a dynamic adjusting novel global harmony search (DANGHS) algorithm, which combines NGHS and dynamic adjustment strategies for genetic mutation probability, is introduced in this paper. Moreover, extensive computational experiments and comparisons are carried out for 14 benchmark continuous optimization problems. The results show that the proposed DANGHS algorithm has better performance in comparison with other HS algorithms in most problems. In addition, the proposed algorithm is more efficient than previous methods. Finally, different strategies are suitable for different situations. Among these strategies, the most interesting and exciting strategy is the periodic dynamic adjustment strategy. For a specific problem, the periodic dynamic adjustment strategy could have better performance in comparison with other decreasing or increasing strategies. These results inspire us to further investigate this kind of periodic dynamic adjustment strategy in future experiments. Full article

This paper concentrates on the detection of steganography in inactive frames of low bit rate audio streams in Voice over Internet Protocol (VoIP) scenarios. Both theoretical and experimental analyses demonstrate that the distribution of 0 and 1 in encoding parameter bits becomes symmetric after a steganographic process. Moreover, this symmetry affects the frequency of each subsequence of parameter bits, and accordingly changes the poker test statistical features of encoding parameter bits. Employing the poker test statistics of each type of encoding parameter bits as detection features, we present a steganalysis method based on a support vector machine. We evaluate the proposed method with a large quantity of speech samples encoded by G.723.1 and compare it with the entropy test. The experimental results show that the proposed method is effective, and largely outperforms the entropy test in any cases. Full article

This paper studies the stress state of a threaded fastening by using Finite Element (FE) models, applied to surgical screws in cortical bone. There is a general interest in studying the stress states induced in the different elements of a joint caused by the thread contact. Analytical models were an initial approach, and later FE models allowed detailed studies of the complex phenomena related to these joints. Different studies have evaluated standard threaded joints in machinery and structures, being the thread symmetric. However, surgical screws employ asymmetric thread geometry, selected to improve the stress level generated in the bone. Despite the interest and widespread use, there is scarce documentation on the actual effect of this thread type. In this work, we discuss the results provided by FE models with detailed descriptions of the contacts comparing differences caused by the materials of the joint, the thread geometry and the thread’s three-dimensional helical effects. The complex contacts at the threaded surfaces cause intense demand on computational resources that often limits the studies including these joints. We analyze the results provided by one commercial software package to simplify the threaded joints. The comparison with detailed FE models allows a definition of the level of uncertainty and possible limitations of this type of simplifications, and helps in making suitable choices for complex applications. Full article

In this article, the authors propose a novel procedure for designing a customized 3D-printed surgical template to guide surgeons in inserting screws into the sacral zone during arthrodesis surgeries. The template is characterized by two cylindrical guides defined by means of trajectories identified, based on standard procedure, via an appropriate Computer-Aided-Design (CAD)-based procedure. The procedure is based on the definition of the insertion direction by means of anatomical landmarks that enable the screws to take advantage of the maximum available bone path. After 3D printing, the template adheres perfectly to the bone surface, showing univocal positioning by exploiting the foramina of the sacrum, great maneuverability due to the presence of an ergonomic handle, as well as a break system for the two independent guides. These features make the product innovative. Thanks to its small size and the easy anchoring, the surgeon can simply position the template on the insertion area and directly insert the screws, without alterations to standard surgical procedures. This has the effect of reducing the overall duration of the surgery and the patient’s exposure to X-rays, and increasing both the safety of the intervention and the quality of the results. Full article

Nowadays, more and more applications are dependent on storage and management of semi-structured information. For scientific research and knowledge-based decision-making, such data often needs to be published, e.g., medical data is released to implement a computer-assisted clinical decision support system. Since this data contains individuals’ privacy, they must be appropriately anonymized before to be released. However, the existing anonymization method based on l-diversity for hierarchical data may cause serious similarity attacks, and cannot protect data privacy very well. In this paper, we utilize fuzzy sets to divide levels for sensitive numerical and categorical attribute values uniformly (a categorical attribute value can be converted into a numerical attribute value according to its frequency of occurrences), and then transform the value levels to sensitivity levels. The privacy model ( ${\alpha }_{lev}^h$ , k)-anonymity for hierarchical data with multi-level sensitivity is proposed. Furthermore, we design a privacy-preserving approach to achieve this privacy model. Experiment results demonstrate that our approach is obviously superior to existing anonymous approach in hierarchical data in terms of utility and security. Full article

Let $G_1$ and $G_2$ be disjoint copies of a graph G and $g:V\left(G_1\right)\to V\left(G_2\right)$ be a function. A functigraph $F_G$ consists of the vertex set $V\left(G_1\right)\cup V\left(G_2\right)$ and the edge set $E\left(G_1\right)\cup E\left(G_2\right)\cup \{uv:g\left(u\right)=v\}$ . In this paper, we extend the study of distinguishing numbers of a graph to its functigraph. We discuss the behavior of distinguishing number in passing from G to $F_G$ and find its sharp lower and upper bounds. We also discuss the distinguishing number of functigraphs of complete graphs and join graphs. Full article

In this paper we provide an application of neutrosophic bipolar fuzzy sets in daily life’s problem related with HOPE foundation that is planning to build a children hospital, which is the main theme of this paper. For it we first develop the theory of neutrosophic bipolar fuzzy sets which is a generalization of bipolar fuzzy sets. After giving the definition we introduce some basic operation of neutrosophic bipolar fuzzy sets and focus on weighted aggregation operators in terms of neutrosophic bipolar fuzzy sets. We define neutrosophic bipolar fuzzy weighted averaging $({\mathcal{N}}^{\mathcal{B}}\mathcal{FWA})$ and neutrosophic bipolar fuzzy ordered weighted averaging $({\mathcal{N}}^{\mathcal{B}}\mathcal{FOWA})$ operators. Next we introduce different kinds of similarity measures of neutrosophic bipolar fuzzy sets. Finally as an application we give an algorithm for the multiple attribute decision making problems under the neutrosophic bipolar fuzzy environment by using the different kinds of neutrosophic bipolar fuzzy weighted/fuzzy ordered weighted aggregation operators with a numerical example related with HOPE foundation. Full article

Linguistic neutrosophic numbers (LNNs) are a powerful tool for describing fuzzy information with three independent linguistic variables (LVs), which express the degrees of truth, uncertainty, and falsity, respectively. However, existing LNNs cannot depict the hesitancy of the decision-maker (DM). To solve this issue, this paper first defines a hesitant linguistic neutrosophic number (HLNN), which consists of a few LNNs regarding an evaluated object due to DMs’ hesitancy to represent their hesitant and uncertain information in the decision-making process. Then, based on the least common multiple cardinality (LCMC), we present generalized distance and similarity measures of HLNNs, and then develop a similarity measure-based multiple-attribute decision-making (MADM) method to handle the MADM problem in the HLNN setting. Finally, the feasibility of the proposed approach is verified by an investment decision case. Full article

Asymmetric windows are of increasing interest to researchers because of the nonlinear and adjustable phase response, as well as alterable time delay. Short-time phase distortion can provide an essential improvement in speech coding, and also has better performance in speech recognition. The merits of asymmetric windows in the aspect of spectral behaviors have an important function in frequency component detection and parameter estimation. In this paper, the phase response of windows were further studied, and the phase characteristics of symmetric and asymmetric windows are described. The relationship between the barycenter of windows in the time domain, and the phase characteristic at the center of the main lobe in the frequency domain, was established. In light of the relationship, an improved version of the asymmetric window- based frequency estimation algorithm was proposed. The improved algorithm has advantages of straightforward implementation and computational efficiency. The numeric simulation results also indicate that the improved approach is more robust than the traditional method against additive random noise. Full article

In this paper, the exact analytical solutions are developed for the thermodynamic behavior of an Euler-Bernoulli beam resting on an elastic foundation and exposed to a time decaying laser pulse that scans over the beam with a uniform velocity. The governing equations, namely the heat conduction equation and the vibration equation are solved using the Green’s function approach. The temporal and special distributions of temperature, deflection, strain, and the energy absorbed by the elastic foundation are calculated. The effects of the laser motion speed, the modulus of elastic foundation reaction, and the laser pulse duration time are studied in detail. Full article

In this paper, we introduce the concept of neutrosophic number from different viewpoints. We define different types of linear and non-linear generalized triangular neutrosophic numbers which are very important for uncertainty theory. We introduced the de-neutrosophication concept for neutrosophic number for triangular neutrosophic numbers. This concept helps us to convert a neutrosophic number into a crisp number. The concepts are followed by two application, namely in imprecise project evaluation review technique and route selection problem. Full article

In this article, we have an explicit description of the binary isosahedral group as a 600-cell. We introduce a method to construct binary polyhedral groups as a subset of quaternions $\mathbb{H}$ via spin map of $SO(3)$. In addition, we show that the binary icosahedral group in $\mathbb{H}$ is the set of vertices of a 600-cell by applying the Coxeter–Dynkin diagram of $H_4$. Full article

Suppose that G is a graph over n vertices. G has n eigenvalues (of adjacency matrix) represented by ${\lambda }_1,{\lambda }_2,\cdots ,{\lambda }_n$. The Gaussian Estrada index, denoted by $H\left(G\right)$ (Estrada et al., Chaos 27(2017) 023109), can be defined as $H\left(G\right)={\sum }_{i=1}^n{e}^{-{\lambda }_i^2}$. Gaussian Estrada index underlines the eigenvalues close to zero, which plays an important role in chemistry reactions, such as molecular stability and molecular magnetic properties. In a network of particles governed by quantum mechanics, this graph-theoretic index is known to account for the information encoded in the eigenvalues of the Hamiltonian near zero by folding the graph spectrum. In this paper, we establish some new lower bounds for $H\left(G\right)$ in terms of the number of vertices, the number of edges, as well as the first Zagreb index. Full article