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In this paper, we focus on the strong product of the pentagonal networks. Let $ R_{n} $ be a hexagonal network composed of $ 2n $ pentagons and $ n $ quadrilaterals. Let $ P_{n}^{2} $ denote the graph formed by the strong product of $ R_{n} $ and its copy $ R_{n}^{\prime} $. By utilizing the decomposition theorem of the normalized Laplacian characteristics polynomial, we characterize the explicit formula of the multiplicative degree-Kirchhoff index completely. Moreover, the complexity of $ P_{n}^{2} $ is determined.

In this paper, we focus on the strong product of the pentagonal networks. Let $ R_{n} $ be a pentagonal network composed of $ 2n $ pentagons and $ n $ quadrilaterals. Let $ P_{n}^{2} $ denote the graph formed by the strong product of $ R_{n} $ and its copy $ R_{n}^{\prime} $. By utilizing the decomposition theorem of the normalized Laplacian characteristics polynomial, we characterize the explicit formula of the multiplicative degree-Kirchhoff index completely. Moreover, the complexity of $ P_{n}^{2} $ is determined.

As a powerful tool for describing and studying the properties of networks, the graph spectrum analyses and calculations have attracted substantial attention from the scientific community. Let $ C_{n} $ represent linear crossed phenylenes. Based on the Laplacian (normalized Laplacian, resp.) polynomial of $ C_{n} $, we first investigated the Laplacian (normalized Laplacian, resp) spectrum of $ C_{n} $ in this paper. Furthermore, the Kirchhoff index, multiplicative degree-Kirchhoff, index and complexity of $ C_{n} $ were obtained through the relationship between the roots and the coefficients of the characteristic polynomials. Finally, it was found that the Kirchhoff index and multiplicative degree-Kirchhoff index of $ C_{n} $ were approximately one quarter of their Wiener index and Gutman index, respectively.

The concept of interval-valued intuitionistic fuzzy sets (IVIFSs) presents a compelling and practical framework for modeling real-world problems. In various fields, such as pattern recognition and decision-making, the development of similarity measures tailored to this class holds significant importance. These measures play a pivotal role in the decision-making process involving IVIFSs, as they quantify the extent of similarity between two such sets. In this article, the shortcomings of the existing similarity measures within the framework of IVIFSs are highlighted, and an improved similarity measure is presented. A comparative study validates that this new similarity measure is better than the existing measures in the IVIF environment. This study systematically establishes several essential properties of the novel similarity measure and substantiates its effectiveness through numerical illustrations. Moreover, a comparative assessment is undertaken to validate the efficacy of the recently introduced measure in relation to established metrics, within the context of IVIFSs. To address the evaluation of software quality, a dedicated mechanism is devised, harnessing the proposed IVIFS similarity measure. Furthermore, an innovative production strategy is formulated utilizing the newly defined methodology to determine the optimal approach for the production of a specific product.

Sylow's theorems are fundamental theorems in classical group theory that are of paramount importance. The extension of these theorems into diverse fuzzy contexts emerges as a compelling area of exploration. This study introduces the novel concept of the conjunctive complex fuzzy conjugate element within the conjunctive complex fuzzy subgroup of a group, elucidating numerous crucial properties of this concept. Additionally, it propounds the notion of the conjunctive complex fuzzy p-subgroup within the conjunctive complex fuzzy subgroup (CCFSG) and delineates various indispensable characteristics associated with this construct. Additionally, the paper formulates the conjunctive complex fuzzy version of the Cauchy theorem for finite groups. Lastly, it defines the concept of the conjunctive complex fuzzy Sylow p-subgroup for a finite group and conducts a generalization of Sylow's theorems within a conjunctive complex fuzzy environment.

IntroductionPathogens causing diabetic foot infections (DFIs) vary by region globally; however, knowledge of the causative organism is essential for effective empirical treatment. We aimed to determine the incidence and antibiotic susceptibility of DFI pathogens worldwide, focusing on Asia and China.MethodsThrough a comprehensive literature search, we identified published studies on organisms isolated from DFI wounds from January 2000 to December 2020.ResultsBased on our inclusion criteria, we analyzed 245 studies that cumulatively reported 38,744 patients and 41,427 isolated microorganisms. DFI pathogens varied according to time and region. Over time, the incidence of Gram-positive and Gram-negative aerobic bacteria have decreased and increased, respectively. America and Asia have the highest (62.74%) and lowest (44.82%) incidence of Gram-negative bacteria, respectively. Africa has the highest incidence (26.90%) of methicillin-resistant Staphylococcus aureus. Asia has the highest incidence (49.36%) of Gram-negative aerobic bacteria with species infection rates as follows: Escherichia coli, 10.77%; Enterobacter spp., 3.95%; and Pseudomonas aeruginosa, 11.08%, with higher local rates in China and Southeast Asia. Linezolid, vancomycin, and teicoplanin were the most active agents against Gram-positive aerobes, while imipenem and cefoperazone-sulbactam were the most active agents against Gram-negative aerobes.DiscussionThis systematic review showed that over 20 years, the pathogens causing DFIs varied considerably over time and region. This data may inform local clinical guidelines on empirical antibiotic therapy for DFI in China and globally. Regular large-scale epidemiological studies are necessary to identify trends in DFI pathogenic bacteria.Systematic review registrationhttps://www.crd.york.ac.uk/prospero/, identifier CRD42023447645.

Due to climate change and human activities, ecological and environmental issues have become increasingly prominent and it is crucial to deeply study the coordinated development between human activities and the ecological environment. Combining panel data from 31 provinces in China spanning from 2011 to 2020, we employed a fixed-effects model, a threshold regression model, and a spatial Durbin model to empirically examine the intricate impacts of population agglomeration on ecological resilience. Our findings indicate that population agglomeration can have an impact on ecological resilience and this impact depends on the combined effects of agglomeration and crowding effects. Also, the impact of population agglomeration on ecological resilience exhibits typical dual-threshold traits due to differences in population size. Furthermore, population agglomeration not only directly impacts the ecological resilience of the local area, but also indirectly affects the ecological resilience of surrounding areas. In conclusion, we have found that population agglomeration does not absolutely impede the development of ecological resilience. On the contrary, to a certain extent, reasonable population agglomeration can even facilitate the progress of ecological resilience.

Quantitative structure-property relationship (QSPR) modeling employs contemporary mathematical tools to predict physicochemical, thermodynamic and biological properties of compounds from their chemical structures. Graphical indices provide those mathematical tools and efficiently correlate physicochemical, thermodynamic and biological properties of compounds. This paper considers the class of benzenoid hydrocarbons (BHs) and investigates the predictive power of commonly occurring graphical temperature indices for determining thermodynamic characteristics of BHs. The entropy (So) & heat capacity (Cp) have been selected to advocate for thermodynamic characteristics. In order to validate the statistical inference, the lower 30 BHs have been opted as test molecules as their experimental data is also publicly available. First, a computer-based method is put forwarded to evaluate temperature indices of a chemical graph. The computational method is, then, employed to compute temperature indices of 30 lower BHs. Certain QPSR models are proposed by utilizing the experimental data of Cp and So for the BHs. Our statistical analysis suggests that the most efficient regression models are, in fact, quadratic. Unexpectedly, some underresearched temperature indices like the general first & second temperature indices perform well having the correlation coefficient >0.9 which is the minimum threshold in a comparative testing. Based on our statistical analysis, three recently proposed temperature indices perform exceptionally well in comparison with all the existing temperature indices. The results suggest their further employability in QSPR modeling. Importantly, our research contributes towards countering proliferation of graphical indices.

Let G=V;E be a simple graph with vertex set V and edge set E. In a graph G, a subset of edges denoted by M is referred to as an edge-dominating set of G if every edge that is not in M is incident to at least one member of M. A set M⊆E is the locating edge-dominating set if for every two edges e1,e2∈E−M, the sets Ne1∩M and Ne2∩M are nonempty and different. The edge domination number γLG of G is the minimum cardinality of all edge-dominating sets of G. The purpose of this study is to determine the locating edge domination number of certain types of claw-free cubic graphs.

The single valued neutrosophic probabilistic hesitant fuzzy rough Einstein aggregation operator (SV-NPHFRE-AO) is an extension of the neutrosophic probabilistic hesitant fuzzy rough set theory. It is a powerful decision-making tool that combines the concepts of neutrosophic logic, probability theory, hesitant fuzzy sets, rough sets, and Einstein aggregation operators. SV-NPHFRE-AO can be applied in many fields, including livestock decision making. Making judgments about a wide range of issues, including feed formulation, breeding program design, disease diagnostics, and market analysis, is part of the process of managing livestock. By combining data from many sources, SV-NPHFRE-AO can assist decision-makers in livestock management in integrating and evaluating diverse criteria, which can result in more informed choices. It also provides a more accurate and comprehensive representation of decision-making problems by considering the multiple criteria involved and the relationships between them. The single valued neutrosophic set (SV-NS) aggregation operators (AOs) based on Einstein properties using hesitant fuzzy sets (HFSs) and probabilistic hesitant fuzzy sets (PHFSs) with rough sets (RSs) are proposed in this study and can handle a large volume of data, making them suitable for complex and large-scale livestock decision-making problems. We first defined SV-neutrosophic probabilistic hesitant fuzzy rough weighted averaging (SV-NPHFRWA), SV-neutrosophic probabilistic hesitant fuzzy rough weighted geometric (SV-NPHFRWG), SV-neutrosophic probabilistic hesitant fuzzy rough ordered weighted averaging (SV-NPHFROWA) and SV-neutrosophic probabilistic hesitant fuzzy rough hybrid weighted averaging (SV-NPHFRHWA) AOs. Then, based on Einstein properties, we extended these operators and developed the single-valued neutrosophic probabilistic hesitant fuzzy rough Einstein weighted averaging (SV-NPHFREWA) operator. Additionally, an illustrative scenario to show the applicability of the suggested decision-making approach is provided, along with a sensitivity analysis and comparison analysis, which demonstrate that its outcomes are realistic and reliable. We also provide another relation between criteria and alternatives of decision-making using neutrosophic information with quaternion context. By using such type of operators, livestock managers can make more informed decisions, leading to better animal health, higher productivity, and increased profitability.

With the rapid development of the economy, the air quality is facing increasingly severe pollution challenges. The air quality is related to public health and the sustainable development of the environment of China. In this paper, we first investigate the changes in the monthly air quality index data of Hefei from 2014 to 2020. Second, we analyze whether the Spring Festival factors lead to the deterioration of the air quality index according to the time sequence. Third, we construct an improved model to predict the air quality index of Hefei. There are three primary discoveries: (1) The air quality index of Hefei has obvious periodicity and a trend of descent. (2) The influencing factors of Spring Festival have no significant effect on the air quality index series. (3) The air quality index of Hefei will maintain a fluctuating and descending trend for a period of time. Finally, some recommendations for the air quality management policy in Hefei are presented based on the obtained results.

The aim of this paper is to illustrate how degree sequences may successfully be used over some graph products. Moreover, by taking into account the degree sequence, we will expose some new distinguishing results on special graph products. We will first consider the degree sequences of tensor and cartesian products of graphs and will obtain the omega invariant of them. After that we will conclude that the set of graphs forms an abelian semigroup in the case of tensor product whereas this same set is actually an abelian monoid in the case of cartesian product. As a consequence of these two operations, we also give a result on distributive law which would be important for future studies.

Based on panel data from 31 provinces in China between 2011 and 2020, we empirically studied the impact of the digital economy on urban resilience using fixed-effects models, threshold-effects models and spatial Durbin models. Our research findings indicate that (1) the development of the digital economy has a significant positive impact on the enhancement of urban resilience; (2) the promotional effect of the digital economy on urban resilience varies significantly across different regions; (3) the promotional effect of the digital economy on urban resilience exhibits a typical double-threshold characteristic due to the different levels of development in digital financial inclusion and (4) the digital economy has a positive spillover effect on the urban resilience of surrounding areas. Therefore, we should actively promote the development of the digital economy and digital financial inclusion, making the digital economy a new driving force for promoting urban resilience.

Cardiovascular disease (CVD) is a leading global health concern. There is a critical need for accurate and reliable decision-making tools to select the optimal approach for diagnosing cardiovascular disease (CVD). In this study, we have addressed this pressing issue. Complex intuitionistic fuzzy set (CIFS) theory is adept at encapsulating vagueness due to its capability to encompass comprehensive problem specifications characterized by both intuitionistic uncertainty and periodicity. Within the scope of this article, we present two novel aggregation operators: the complex intuitionistic fuzzy dynamic weighted averaging (CIFDWA) operator and the complex intuitionistic fuzzy dynamic weighted geometric (CIFDWG) operator. Some intriguing characteristics of these operators are elucidated, and important special cases are also defined in detail. We devise an enhanced score function to rectify the deficiencies observed in the existing score function under complex intuitionistic fuzzy knowledge. Furthermore, these operators are employed in the development of a systematic approach for the handling of multiple attribute decision-making (MADM) scenarios involving complex intuitionistic fuzzy data. Moreover, we undertake the resolution of an MADM problem, wherein we ascertain the optimal approach for diagnosing cardiovascular disease (CVD) through the utilization of the proposed operators, thereby substantiating their utility in decision-making processes. Finally, we conduct a comprehensive comparative analysis, pitting the presented operators against an array of existing counterparts, in order to demonstrate the reliability and stability inherent in the derived methodologies.

Deep Neural Networks and Optimization Algorithms have many applications in engineering problems and scientific research [...]

We present new Mercer variants of Hermite-Hadamard (HH) type inequalities via Atangana-Baleanu (AB) fractional integral operators pertaining non-local and non-singular kernels. We establish trapezoidal type identities for fractional operator involving non-singular kernel and give Jensen-Mercer (JM) variants of Hermite-Hadamard type inequalities for differentiable mapping Υ possessing convex absolute derivatives. We establish connections of our results with several renowned results in the literature and also give applications to special functions.

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 PECn with the help of graph theory. Based on the networks PECn, 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.

Two graphs are said to be cospectral with respect to the Laplacian matrix if they have the same Laplacian spectrum. A graph is said to be determined by the Laplacian spectrum if there is no other non-isomorphic graph with the same Laplacian spectrum. In this paper, we prove that one special class of triple starlike tree is determined by its Laplacian spectrum.

In this work, we incorporate modular arithmetic and discuss a special class of graphs based on power functions in a given modulus, called power digraphs. In power digraphs, the study of cyclic structures and enumeration of components is a difficult task. In this manuscript, we attempt to solve the problem for $p$th power congruences over different classes of residues, where $p$ is an odd prime. For any positive integer $m$, we build a digraph $G(p,m)$ whose vertex set is $\mathbb{Z}_{m} = \{0, 1, 2, 3,..., m-1\}$ and there will be a directed edge from vertices $u\in \mathbb{Z}_{m}$ to $v\in \mathbb{Z}_{m} $ if and only if $u^{p}\equiv v~ (\textmd{mod} ~m)$. We study the structures of $G(p,m)$. For the classes of numbers $2^{r}$ and $p^{r}$ where $r\in \mathbb{Z^{+}}$, we classify cyclic vertices and enumerate components of $G(p,m)$. Additionally, we investigate two induced subdigraphs of $G(p,m)$ whose vertices are coprime to $m$ and not coprime to $m$, respectively. Finally, we characterize regularity and semiregularity of $G(p,m)$ and establish some necessary conditions for cyclicity of $G(p,m)$.

The resistance distance is an intrinsic metric on graphs that have been extensively studied by many physicists and mathematicians. The resistance distance between two vertices of a simple connected graph $G$ is equal to the resistance between two equivalent points on an electrical network, constructed to correspond to $G$ , with each edge being replaced by a unit resistor. Hypercube $Q_{n}$ is one of the most efficient and versatile topological structures of the interconnection networks, which received much attention over the past few years. The folded $n$ -cube graph is obtained from hypercube $Q_{n}$ by merging vertices of the hypercube $Q_{n}$ that are antipodal, i.e., lie at a distance $n$ . Folded $n$ -cube graphs have been studied in parallel computing as a potential network topology. The folded $n$ -cube has the same number of vertices but half the diameter as compared to hypercubes which play an important role in analyzing the efficiency of interconnection networks. We intend is to minimize the diameter. In this study, we will compute the resistance distance between any two vertices of the folded $n$ -cube by using the symmetry method and classic Kirchhoff’s equations. This method is beneficial for distance-transitive graphs. As an application, we will also give an example and compute the resistance distance in the Biggs-Smith graph, which shows the competency of the proposed method.

Based on the Chinese General Social Survey database (2010–2015), this article explores the relationship between income inequality and residents’ subjective well-being from the perspective of inequality of opportunity and inequality of effort. We find that inequality of opportunity has a negative impact on subjective well-being in China, where inequality of effort has a positive impact. Our empirical results are robust for changing the inequality indicators. In the sub-sample studies, consistent conclusions are obtained in rural areas, whereas in urban areas only inequality of effort has a significant impact. The results of mechanism study show that inequality of opportunity decreases residents’ sense of fairness, and inequality of effort increases residents’ sense of fairness, thus affecting their subjective well-being. The results of this study provide a good response to the inconclusive research findings on the impact of income inequality on subjective well-being.

Energies and spectrum of graphs associated to different linear operators play a significant role in molecular chemistry, polymerisation, pharmacy, computer networking and communication systems. In current article, we compute closed forms of signless Laplacian and Laplacian spectra and energies of multi-step wheel networks Wn, m. These wheel networks are useful in networking and communication, as every node is one hoop neighbour to other. We also present our results for wheel graphs as particular cases. In the end, correlation of these energies on the involved parameters m ≥ 3 and n is given graphically. Present results are the natural generalizations of the already available results in the literature.

Abstract In this paper, we introduce the generalized K $\mathcal{K}$ -fractional integral in the frame of a new parameter K > 0 $\mathcal{K}>0$ . This paper offers some new important inequalities of Grüss type using the generalized K $\mathcal{K}$ -fractional integral and associated integral inequalities. Our results with this new integral operator have the abilities to be implemented for the evaluation of many mathematical problems related to the real world applications.

Mathematical modeling, coding or labeling with the help of numeric numbers based on the parameter of distance plays a vital role in the studies of the structural properties of the networks such as accessibility, centrality, clustering, complexity, connectivity, modularity, robustness and vulnerability. In particular, various distance based dimensions of the networks are used to rectify the problems in different strata of computer science and chemistry such as navigation, image processing, pattern recognition, integer programming problem, drug discovery and formation of different chemical compounds. In this note, we consider a family of rotationally symmetric and planar networks called by circular ladders consisting of different faced triangles, quadrangles and pentagons. We compute local fractional metric dimensions of the aforesaid networks and study their boundedness. Moreover, our findings at the closure of this note have been summarized in the form of tables and 3-D plots.

A vertex u ∈ V(G) resolves (distinguish or recognize) two elements (vertices or edges) v, w ∈ E(G)UV(G) if dG(u, v) ≠ dG(u, w) . A subset Lm of vertices in a connected graph G is called a mixed metric generator for G if every two distinct elements (vertices and edges) of G are resolved by some vertex set of Lm. The minimum cardinality of a mixed metric generator for G is called the mixed metric dimension and is denoted by dimm(G). In this paper, we studied the mixed metric dimension for three families of graphs Dn, An, and Rn, known from the literature. We proved that, for Dn the dimm(Dn) = dime(Dn) = dim(Dn), when n is even, and for An the dimm(An) = dime(An), when n is even and odd. The graph Rn has mixed metric dimension 5.

The universality of M-polynomial paves way towards establishing closed forms of many leading degree-based topological indices as it is done by Hosoya polynomial for distance-based indices. The study of topological indices is recently one of the most active research areas in chemical graph theory. The aim of this paper is to establish closed formulas for M-polynomials of Linear chains of benzene, napthalene, and anthracene graphs. From this polynomial we also compute as many as nine degree-based topological indices for these three chains. Our results will potentially play an important role in pharmacy, drug design, and many other applied areas of molecular sciences.

Dual hesitant fuzzy sets (DHFSs) is the refinement and extension of hesitant fuzzy sets and encompasses fuzzy sets, intuitionistic fuzzy sets, hesitant fuzzy sets, and fuzzy multisets as a special case. DHFSs have two parts, that is, the membership function and the non-membership function, in which each function is defined by two sets of some feasible values. Therefore, according to the practical demand, DHFSs are more adjustable than the existing ones and provide the information regarding different objects in much better way. The set pair analysis (SPA) illustrates unsureness in three angles, called “identity”, “discrepancy” and “contrary”, and the connection number (CN) is one of its main features. In the present article, the axiom definition of distance measure between DHFSs and CN is introduced. The distance measures are established on the basis of Hamming distance, Hausdorff distance and Euclidean distance. The previous identities and relationship between them are discussed in detail. On the basis of the geometric distance model, the set-theoretic approach, and the matching functions several novel distance formulas of CN are introduced. The novel distance formulas are then applied to multiple-attribute decision making for dual hesitant fuzzy environments. Finally, to demonstrate the validity of the introduced measures, a practical example of decision-making is presented. The benefits of the new measures over the past measures are additionally talked about.

Metric dimension is a distance based parameter which is used to determine the locations of machines (or robots) with respect to minimum consumption of time, shortest distance among the destinations and lesser number of the utilized nodes as places of the objects. It is also used to characterize the chemical compounds in the molecular networks in the form of their unique presentations. These are problems worth investigating in different strata of computer science and chemistry such as navigation, combinatorial optimization, pattern recognition, image processing, integer programming, network theory and drugs discovery. In this paper, a general computational criteria is established to compute the local fractional metric dimension (LFMD) of connected networks in the form of sharp lower and upper bounds. A complete characterization of the connected networks whose LFMDs attain the exactly lower bound is obtained and some particular classes of networks (complete networks, generalized windmill and $h$ -level windmill) whose LFMDs attain the exactly upper bound are also addressed. In the consequence of the main obtained criteria, LFMDs of wheel-related networks (anti-web gear, $m$ -level wheel, prism, helm and flower) are computed and their boundedness (or un-boundedness) is also illustrated with the help of 2D and 3D graphical presentations.

With the popularization of higher education and the promotion of college enrollment expansion, the number of college graduates increases sharply. At the same time, the continuous transformation and upgrading of the industrial structure put forward higher requirements on the employability of college students, which leads to the imbalance between supply and demand in the labor market. The key to dealing with employment difficulties lie in the improvement of college students’ employability. Therefore, we make a regression analysis of 263 valid samples from universities in Anhui Province and extract the factors that influence the improvement of college students’ employability in the process of talent cultivation in university. The result shows that there is a positive correlation between course setting, course teaching, club activities, and college students’ employability, among which the course teaching and club activities are the most critical factors which may influence college students’ employability. In addition, from the viewpoint of individual college students, the overall grades of college students and the time of participating in the internship are also closely related to their employability, i.e., college students with good overall grades and long internship time should also have stronger employability.

The irregularities of graphs and their dependencies on the size parameters are recently attracting attention of not only mathematician but also of theoretical chemists. It is found that these irregularities are related with the properties of the substance involved. Cerium oxide is a rare-earth metal having formula CeO2 and it is light yellow-white powder. In the present article we are concerned with computing the closed forms of irregularity measures of general form of crystal structure of Cerium Oxide CeO2 based on mathematical model and calculation.

In order to fully implement the new development concept, bring into full play the potential of sports development, and maintain the resilience of China’s sports development. This paper studies the resilience evaluation and spatial correlation of Chinese sports development under the new development concept. First, we constructed Resilience Evaluation Indexes System for Sports Development in China based on the analysis of the resilience features of sports development and the DPSIR model, which is from the five aspects of “driving force – pressure – state – influence – response.” Second, used Coefficient of Variation and Technique for Order Preference by Similarity to an Ideal Solution (TOPSIS) Method to measure the resilience level of sports development in 31 provinces in China from 2013 to 2017. Then, we introduced the obstacle degree model to identify the obstacle factors that hinder the resilience of Chinese sports development in different periods. Finally, we used the global and local Moran indexes to analyze the spatial correlation of China sports regional development. The results showed that: (1) overall, the development level of sports resilience in 31 provinces in China showed an upward trend from 2013 to 2017, while some provinces showed obvious fluctuations. (2) The obstacles to the development of sports resilience in China mainly include sports scientific research equipment, the number of national fitness monitoring stations, the number of national fitness centers, the full-time equivalent of (R&D) personnel, and the number of sports scientific research projects. The response subsystem is the main obstacle factor that affects the improvement of the resilience level of sports development in China. (3) There is a positive spatial autocorrelation between the resilience level of sports development and regional spatial distribution, and the correlation shows a weakening trend, and the internal difference is significant. Finally, we concluded that we must take the new development philosophy as the guiding principle. First, we should stick to innovation-driven development to fully upgrade the resilience of China’s sports development. Second, we should adhere to the principle of coordinated development to promote the overall and balanced development of sports. Lastly, we should promote shared development so as to deliver benefits for all in an equal way.

The degree distance index (DDI) is a vertex-degree weighted version of a well-known index that is called by Wiener index (WI). In extremal theory of graphs, improving the bounds with best possible values is a worth investigating problem. In this note, the exact formulae of the DDI for the four different types of the sum graphs in the terms of various indices of their factor graphs are computed. Moreover, a comparison is also presented between the obtained exact and already existing bounded values for the particular sum graphs.

Resolving parameters are a fundamental area of combinatorics with applications not only to many branches of combinatorics but also to other sciences. In this study, we construct a class of Toeplitz graphs and will be denoted by T2nW so that they are Cayley graphs. First, we review some of the features of this class of graphs. In fact, this class of graphs is 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 because of the difficulty of the computing resolving parameters of a class of graphs which are not distance regular, 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.

The aim of this work is the reduction of the throttling noise when the capillary is used as a throttling device. Based on the theory of bubble dynamics, two-phase flow, and aerated supercavitation, four different sizes of aerated devices used in refrigerator refrigeration systems are designed. Throttling noise and the temperature and pressure of inlet and outlet of the capillary are measured under stable operation. To compare the noise suppression effects in different groups of experiments, we introduced the cavitation number to analyze, revealed the principle of aerated supercavitation to suppress noise, and combined the results of Fluent simulations to get the relationship between the noise suppression effect and the aerated quality. The experimental results showed that the aerated device can obviously suppress the throttling noise of the capillary outlet, up to 2.63 dB(A), which provides a new way for reducing the capillary throttling noise.

For a graph G, its sum-connectivity index is denoted by χG and is defined as the sum of the numbers du+dv−1/2 over all edges uv of G, where dw denotes the degree of a vertex w∈VG. In this study, we find a sharp lower bound on the sum-connectivity index of graphs having minimum degree of at least 3 under certain constraints and characterize the corresponding extremal graphs.

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

It is well known that many topological indices have widespread use in lots of fields about scientific research, and the Kirchhoff index plays a major role in many different sectors over the years. Recently, Li et al. (Appl. Math. Comput. 382 (2020) 125335) proposed the problem of determining the Kirchhoff index and multiplicative degree-Kirchhoff index of graphs derived from Sn×K2, the Cartesian product of the star Sn, and the complete graph K2. In the present study, we completely solve this problem, that is, the explicit closed-form formulae of the Kirchhoff index, multiplicative degree-Kirchhoff index, and number of spanning trees are obtained for some graphs derived from Sn×K2.

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.

Hex-derived network has an assortment of significant applications in medicine store, equipment, and network organization. Graph entropy depends upon distribution probability of vertex set and on graph itself. There are numerous issues in discrete math, software engineering, statistics, and data innovation where graph entropies are utilized to portray the reasonable constructions. In this paper, we talk about hex-derived network of type 3 denoted as HDN3n. We likewise figure degree-based entropies, for example, Randic’, ABC, and GA entropy of HDN3n.

Entropy is a measure of a system’s molecular disorder or unpredictability since work is produced by organized molecular motion. Shannon’s entropy metric is applied to represent a random graph’s variability. Entropy is a thermodynamic function in physics that, based on the variety of possible configurations for molecules to take, describes the randomness and disorder of molecules in a given system or process. Numerous issues in the fields of mathematics, biology, chemical graph theory, organic and inorganic chemistry, and other disciplines are resolved using distance-based entropy. These applications cover quantifying molecules’ chemical and electrical structures, signal processing, structural investigations on crystals, and molecular ensembles. In this paper, we look at K-Banhatti entropies using K-Banhatti indices for C6H6 embedded in different chemical networks. Our goal is to investigate the valency-based molecular invariants and K-Banhatti entropies for three chemical networks: the circumnaphthalene (CNBn), the honeycomb (HBn), and the pyrene (PYn). In order to reach conclusions, we apply the method of atom-bond partitioning based on valences, which is an application of spectral graph theory. We obtain the precise values of the first K-Banhatti entropy, the second K-Banhatti entropy, the first hyper K-Banhatti entropy, and the second hyper K-Banhatti entropy for the three chemical networks in the main results and conclusion.

A topological index as a graph parameter was obtained mathematically from the graph’s topological structure. These indices are useful for measuring the various chemical characteristics of chemical compounds in the chemical graph theory. The number of atoms that surround an atom in the molecular structure of a chemical compound determines its valency. A significant number of valency-based molecular invariants have been proposed, which connect various physicochemical aspects of chemical compounds, such as vapour pressure, stability, elastic energy, and numerous others. Molecules are linked with numerical values in a molecular network, and topological indices are a term for these values. In theoretical chemistry, topological indices are frequently used to simulate the physicochemical characteristics of chemical molecules. Zagreb indices are commonly employed by mathematicians to determine the strain energy, melting point, boiling temperature, distortion, and stability of a chemical compound. The purpose of this study is to look at valency-based molecular invariants for SiO4 embedded in a silicate chain under various conditions. To obtain the outcomes, the approach of atom–bond partitioning according to atom valences was applied by using the application of spectral graph theory, and we obtained different tables of atom—bond partitions of SiO4. We obtained exact values of valency-based molecular invariants, notably the first Zagreb, the second Zagreb, the hyper-Zagreb, the modified Zagreb, the enhanced Zagreb, and the redefined Zagreb (first, second, and third). We also provide a graphical depiction of the results that explains the reliance of topological indices on the specified polynomial structure parameters.

Entropy is a thermodynamic function in chemistry that reflects the randomness and disorder of molecules in a particular system or process based on the number of alternative configurations accessible to them. Distance-based entropy is used to solve a variety of difficulties in biology, chemical graph theory, organic and inorganic chemistry, and other fields. In this article, the characterization of the crystal structure of niobium oxide and a metal–organic framework is investigated. We also use the information function to compute entropies by building these structures with degree-based indices including the K-Banhatti indices, the first redefined Zagreb index, the second redefined Zagreb index, the third redefined Zagreb index, and the atom-bond sum connectivity index.

The numerical coding of the molecular structures on the bases of topological indices plays an important role in the subject of Cheminformatics which is a combination of Computer, Chemistry, and Information Science. In 1972, it was shown that the total π-electron energy of a molecular graph depends upon its structure and it can be obtained by the sum of the square of degrees of the vertices of a molecular graph. Later on, this sum was named as the first Zagreb index. In 2005, for γεR - {0, 1}, the first general Zagreb index of a graph G is defined as Mγ(G) = ΣvεV(G)[dG(v)]γ, where dG(v) is degree of the vertex v in G. In this paper, for each γεR - {0, 1}, we study the first general Zagreb index of the cartesian product of two graphs such that one of the graphs is D-sum graph and the other is any connected graph, where D-sum graph is obtained by using certain D operations on a connected graph. The obtained results are general extensions of the results of Deng et al. [Applied Mathematics and Computation 275(2016): 422-431] and Akhter et al. [AKCE Int. J. Graphs Combin. 14(2017): 70-79] who proved only for γ = 2 and γ = 3, respectively.

Quantitative structure-activity and structure-property associations of natural systems require terminologies for their topological properties. Structure-based topological descriptors/indices of these systems allow these chemical possessions and the bioactivities of these compounds through reckonable structure-activity and structure-property associations' procedures. In this paper, we propose a computational technique to compute analytically exact expressions for certain degree and distance-based topological indices for general graphs. A comparative analysis is conducted with the known techniques where certain experiments are performed to show that our technique is more efficient and possesses less algorithmic and computational complexity. We apply our method to compute explicit expressions of certain degree and distance topological indices for certain infinite families of fullerenes, carbon nanotubes, and carbon nanocones. The obtained results in this paper generalize certain known results in the literature.

The position of a moving point in a connected graph can be identified by computing the distance from the point to a set of sonar stations which have been appropriately situated in the graph. Let Q = {q1, q2, ... , qk} be an ordered set of vertices of a graph G and a is any vertex in G, then the code/representation of a w.r.t Q is the k-tuple (r(a, q1), r(a, q2), ... , r(a, qk)), denoted by r(a|Q). If the different vertices of G have the different representations w.r.t Q, then Q is known as a resolving set/locating set. A resolving/locating set having the least number of vertices is the basis for G and the number of vertices in the basis is called metric dimension of G and it is represented as dim(G). In this paper, the metric dimension of Toeplitz graphs generated by two and three parameters denoted by Tn〈1, t〉 and Tn〈1, 2, t〉, respectively is discussed and proved that it is constant.

Graph labeling has wide applications in the field of computer science, such as coding theory, cryptography, software testing, database management systems, computer architecture, and networking. The computers connected in a network can now be converted in a graph and labels assigned to the graph so formed will help to regulate bandwidth, data traffic, in coding and decoding signals. Let A = (V(Λ), E(Λ)) beagraphwith|V(Λ)| = m and|V(Λ)| = n.Abijection from ζ : V(Λ)∪E(Λ) → {1, 2, 3, ⋯,m + n} is called (a, d)-edge antimagic total labeling if the edge-weights ζ(x) + ζ(xy) + ζ(y) for each xy ∈ E(Λ) form a sequence of consecutive positive integers with minimum edge-weight a and common difference d. In addition, it is called super (a, d)-edge antimagic total labeling if vertices receive the smallest labels. Enomoto et al. (2000) proposed the conjecture that every tree admits super (a, 0)-EAT labeling. In this note, bounds of the minimum and maximum edge-weights for super (a, d)-EAT labeling on the more generalized class of subdivided caterpillars are obtained. Moreover, we have investigated the existence of super (a, d)EAT labeling for the validation of the obtained bounds and the partial support of the aforesaid conjecture, where d ∈ {0, 1, 2}. In fact, the obtained results are a general extension of the results Akhlaq et al. [Utiltas Mathematica, 98 (2015), 227-249].

The degradation data have been widely applied in reliability analysis for deteriorating systems. However, the degradation data are usually contaminated by measurement errors that can severely affect the life estimation performance. This paper presents a degradation modeling and life estimation method using a Wiener degradation model by taking temporal uncertainty, measurement uncertainty, and unit-to-unit heterogeneity into account. The truncated normal distribution is employed to characterize the unit-to-unit heterogeneity in a population due to the fact that the degradation rates of many systems often manifest as positive values. The exact and explicit expressions of the life distribution are derived in the sense of the first hitting time by considering three kinds of uncertainties. The expectation maximization algorithm is used to estimate the model parameters because the resulting likelihood function includes hidden variables, which improves the estimation efficiency compared with the existing maximum likelihood estimation procedure. The effectiveness of the proposed approach is validated through a simulation example and a case study involving the degradation dataset of the LED.

Let Q(G) be the graph derived from G by inserting a new vertex into every edge of G and by connecting edges of these new vertices that are on adjacent edges of G, and Q-double join of G, G1 and G2 denoted by GQ v{G1, G2}, is the graph derived from Q(G), G1 and G2, by connecting every old-vertex V (G) of Q(G) with every vertex of G1 and every new-vertex I(G) of Q(G) with every vertex of G2. In this paper, we mainly considered the resistance distance and Kirchhoff index of Q-double join graphs GQ v {G1, G2} of an r-regular graph G and any two graphs of G1 and G2.

Analysts 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 airship wings and different structures. Topological indices are numerical parameters of a molecular graph which characterize its topology and are usually graph invariant. Topological indices are used, for example, in the development of quantitative structure-activity relationships (QSARs) in which the biological activity or other properties of molecules are correlated with their chemical structure. Topological indices catch symmetry of molecular structures and help us to predict properties, for example, boiling points, viscosity, and the radius of gyrations of nanotubes. In this paper, we compute ${M}$ -polynomials of two nanotubes, $VC_{5} C_{7} [p,q]$ and $HC_{5} C_{7} [p,q]$ . By applying calculus on these ${M}$ -polynomials, we produce formulas of numerous degree-based topological indices, which are functions relying on parameters of the structure and, in combination, decide properties of the concerned polymeric compound.