This paper proposes a user electricity data mining method based on deep learning and improved locust optimization algorithm, and at the same time adopts the Pearson correlation coefficient method to reduce its dimension to improve the data mining effect of linear weighted KFCM algorithm. In order to deal with the electricity demand of massive electricity customers, the user electricity demand forecasting model is constructed based on the Extreme Learning Machine ELM algorithm by combining the relationship between short-term loads and factors of electricity customers. Construct the service optimization model with the maximization of benefit index as the objective function, and use the BAS algorithm to solve the optimal solution in order to achieve the effect of user service optimization. Determine the experimental platform and model parameters, and carry out an example analysis of demand forecasting and service optimization for electricity users.C class users have a small electricity load except for breakfast and dinner, and the maximum time period of the electricity load is from 18:00 to 20:00 hours. Combined with MAPE, the ELM model improves 4.57% than SVR, 21.9% than LSTM, and 34.37% than ARIMA, which indicates that the ELM model is more effective and higher in demand forecasting for electricity users. In addition, the optimal solution of the effect of the BAS algorithm is 69 yuan, 102 yuan and 49 yuan higher than that of the GA algorithm in terms of dividend transmission benefit, energy saving and emission reduction benefit, and electricity right trading benefit, respectively, and the optimal solution based on the BAS algorithm is closer to the actual benefit value, which fully proves the effectiveness of the service optimization model based on the BAS algorithm.
Smart contract technology based on artificial intelligence background is gradually becoming a brand-new path to improve the efficiency of economic transactions due to its unique advantages. This paper initially explores the impact of smart contract technology on economic transaction efficiency through empirical analysis of models and data. The credit mechanism is introduced as an intermediate variable to analyze its mediating effect in the process of improving economic transaction efficiency by smart contract technology. The optimization of Fabric transaction mechanism is realized by using the improved credit model, which further exerts the role of smart contract technology in enhancing economic transaction efficiency. The principal component analysis is used to calculate the comprehensive score of economic transaction efficiency before and after the optimization of smart contract trading mechanism to show the effect of the development of smart contract technology on the improvement of economic transaction efficiency. This paper concludes that the development of smart contract technology will significantly and positively promote the improvement of economic transaction efficiency through benchmark regression analysis, mediation effect test and other methods. After the optimization of smart contract transaction mechanism, the comprehensive score of economic transaction efficiency produces significant improvement compared with the pre-optimization period, in which the average value of the comprehensive score of transaction efficiency in Guangdong, Jiangsu, Shanghai, and Beijing is improved by 20.18%, 24.52%, 33.77%, and 35.54%, respectively. It further indicates that smart contract technology is an effective path to improve economic transaction efficiency.
With the development of renewable energy technology and the pursuit of sustainable development in the construction industry, the design of direct-soft photovoltaic systems integrated with buildings has become an important research direction. In this paper, a variety of photovoltaic power generation modules are selected and combined with building roof functions to design a solar photovoltaic building integration system. In addition, this paper constructs a multi-objective optimization configuration model, improves the multi-objective particle swarm algorithm, and analyzes the optimization effect of the improved particle swarm algorithm on the photovoltaic building integration system by using multiple sets of test functions and evaluation indexes, combined with a number of experiments. The improved particle swarm algorithm in this paper converges to the optimal value of 0.21 when iterating to 25 rounds. And with the increase of the number of nodes, the optimized particle swarm algorithm, the distribution of node voltages in the vicinity of the standard voltage. The PV building integrated system designed in this paper still has a generation output efficiency higher than 85% after 20 years, which shows good stability of power generation. And the power generation in its whole life cycle is about 1645710kwh, which greatly reduces the consumption of conventional energy. In conclusion, the PV building integrated system in this paper not only has significant advantages in terms of capacity efficiency, but also shows strong potential for environmental protection.
As the birthplace of national culture, traditional villages can convey cultural and social natures through spatial configuration. Based on the theory of spatial syntax, this paper combines the genetic algorithm to design the fitness function for optimization, and selects the streets and lanes of Wengji Village as the research sample, focusing on the analysis of its morphological evolution mechanism from 1975 to 2020. Through quantitative analysis, it is found that although the streets and alleys of Wengji Village show spatial scale expansion due to social and economic development, the village streets and alleys can still maintain the original spatial texture and style. The integration degree, selectability, synergy (0.4273~0.6395) and comprehensibility (0.3744~0.5761) of the streets and alleys in Wengji Village are all characterized by increasing, indicating that the spatial accessibility, spatial openness and spatial wholeness of the streets and alleys in Wengji Village have been improved. However, the degree of synergy and comprehensibility are still lower than 0.7, and there is some room for optimization of the wholeness and cognizability of the streets and lanes of Wengji Village. It is necessary to protect and continue the overall structure of the village, optimize and integrate the key spaces of the village, and rationally control the development process of the village, so as to promote the protection of the spatial form of the streets and lanes of Wengji Village and the continuation of the cultural lineage.
In the context of big data, many problems connected with usage of classical techniques in computing large and complicated data have emerged. For a true disruption of silicon and computational power, there is quantum computing which is poised to redefine itself in the future. Quantum computing is the area of study for this research; real-time big data processing with a specific emphasis on speed. In a sequence of experiments, the performance of quantum algorithms was compared to the performance of classical algorithms regarding large-scale data processes. These findings imply that the compared quantum algorithm has enhanced the processing time while providing on average 70%-time reduction as well as 30 percent improvement of the computational efficiency. Besides, the scalability of quantum computing is also better; it remains effective when used on large datasets, and it has an accuracy of 95% that is higher than classical one. This outcome depiction the power of quantum computing in the alteration of data processing tactics. However, there are some issues like quantum decoherence and error rates, which are guaranteeing the non-stability of quantum computing, but they are expecting more improvements in the field of quantum hardware and error correction. From this study, it consequently becomes clear why the advancement of quantum technology should continue and for what; to meet the challenges of big data.
Behera and Panda defined a balancing number as a number b for which the sum of the numbers from \(1\) to \(b – 1\) is equal to the sum of the numbers from \(b + 1\) to \(b + r\) for some r. They also classified all such numbers. We define two notions of balancing numbers for Farey fractions and enumerate all possible solutions. In the stricter definition, there is exactly one solution, whereas in the weaker one all sufficiently large numbers work. We also define notions of balancing numbers for levers and mobiles, then show that these variants have many acceptable arrangements. For an arbitrary mobile, we prove that we can place disjoint consecutive sequences at each of the leaves and still have the mobile balance. However, if we impose certain additional restrictions, then it is impossible to balance a mobile.
For a graph G and for non-negative integers p, q, and r, the triplet \((p, q, r)\) is said to be an admissible triplet if \(3p + 4q + 6r = |E(G)|\). If G admits a decomposition into p cycles of length 3, q cycles of length 4, and r cycles of length 6 for every admissible triplet \((p, q, r)\), then we say that G has a \(\{C_{3}^{p}, C_{4}^{q}, C_{6}^{r}\}\)-decomposition. In this paper, the necessary conditions for the existence of \(\{C_{3}^{p}, C_{4}^{q}, C_{6}^{r}\}\)-decomposition of \(K_{\ell, m, n} (\ell \leq m \leq n)\) are proved to be sufficient. This affirmatively answers the problem raised in Decomposing complete tripartite graphs into cycles of lengths 3 and 4, Discrete Math. 197/198 (1999), 123-135. As a corollary, we deduce the main results of Decomposing complete tripartite graphs into cycles of lengths 3 and 4, Discrete Math., 197/198, 123-135 (1999) and Decompositions of complete tripartite graphs into cycles of lengths 3 and 6, Austral. J. Combin., 73(1), 220-241 (2019).
A graph G(V, E) is Γ-harmonious when there is an injection f from V to an Abelian group Γ such that the induced edge labels defined as w(xy) = f(x) + f(y) form a bijection from E to Γ. We study Γ-harmonious labelings of several cycles-related classes of graphs, including Dutch windmills, generalized prisms, generalized closed and open webs, and superwheels.
If Γ is a finite group and G a graph such that Aut(G) ≡ Γ acts regularly on V(G), then we say that G is a graphical regular representation (GRR) of Γ. The question asking which finite groups have at least one GRR was an important question in algebraic graph theory and it was completely solved as a result of work done by several researchers. However, it remains a challenge to discern whether a group known to have GRRs has GRRs with specific properties, such as being trivalent. In this paper, we shall be deriving simple conditions on the parameters of a subset of a dihedral group for easily constructing trivalent graphical regular representations (GRR) of the group. Specifically, we shall prove the following:
Let n be an odd integer greater than 5 and let r, s, and t be integers less than n such that the difference of any two of them is relatively prime to n. If 3r – 2s = t (mod n), then Cay(Dn, {abr, abs, abt}) is a GRR of Dn.
We will also be looking at very convenient corollaries of this result. But another main aim of this paper is to show how a simple use of Schur rings can be used to derive such results. This paper therefore also serves as a review of some basic results about Schur rings which we feel should be among the standard armory of an algebraic graph theorist.
This paper presents an investigation of a modified Leslie-Gower predator-prey model that incorporates fractional discrete-time Michaelis-Menten type prey harvesting. The analysis focuses on the topology of nonnegative interior fixed points, including their existence and stability dynamics. We derive conditions for the occurrence of flip and Neimark-Sacker bifurcations using the center manifold theorem and bifurcation theory. Numerical simulations, conducted with a computer package, are presented to demonstrate the consistency of the theoretical findings. Overall, our study sheds light on the complex dynamics that arise in this model and highlights the importance of considering fractional calculus in predator-prey systems with harvesting.
For a set \( S \) of vertices in a connected graph \( G \), the multiplicative distance of a vertex \( v \) with respect to \( S \) is defined by \(d_{S}^{*}(v) = \prod\limits_{x \in S, x \neq v} d(v,x).\) If \( d_{S}^{*}(u) \neq d_{S}^{*}(v) \) for each pair \( u,v \) of distinct vertices of \( G \), then \( S \) is called a multiplicative distance-locating set of \( G \). The minimum cardinality of a multiplicative distance-locating set of \( G \) is called its multiplicative distance-location number \( loc_{d}^{*}(G) \). If \( d_{S}^{*}(u) \neq d_{S}^{*}(v) \) for each pair \( u,v \) of distinct vertices of \( G-S \), then \( S \) is called an external multiplicative distance-locating set of \( G \). The minimum cardinality of an external multiplicative distance-locating set of \( G \) is called its external multiplicative location number \( loc_{e}^{*}(G) \). We prove the existence or non-existence of multiplicative distance-locating sets in some well-known classes of connected graphs. Also, we introduce a family of connected graphs such that \( loc_{d}^{*}(G) \) exists. Moreover, there are infinite classes of connected graphs \( G \) for which \( loc_{d}^{*}(G) \) exists as well as infinite classes of connected graphs \( G \) for which \( loc_{d}^{*}(G) \) does not exist. A lower bound for the multiplicative distance-location number of a connected graph is established in terms of its order and diameter.
Earlier optimal key pre\(-\)distribution schemes (KPSs) for distributed sensor networks (DSNs) were proposed using combinatorial designs via transversal designs, affine, and partially affine resolvable designs. Here, nearly optimal KPSs are introduced and a class of such KPSs is obtained from resolvable group divisible designs. These KPSs are nearly optimal in the sense of local connectivity. A metric for the efficiency of KPSs is given. Further, an optimal KPS has also been proposed using affine resolvable \( L_{2} \)-type design.
We study the nonzero algebraic real algebras \( A \) with no nonzero joint divisor of zero. We prove that if \( Z(A) \neq 0 \) and \( A \) satisfies one of the Moufang identities, then \( A \) is isomorphic to \( \mathbb{R} \), \( \mathbb{C} \), \( \mathbb{H} \), or \( \mathbb{O} \). We show also that if \( A \) is power-associative, flexible, and satisfies the identity \( (a,a,[a,b])=0 \), then \( A \) is isomorphic to \( \mathbb{R} \), \( \mathbb{C} \), \( \mathbb{H} \), or \( \mathbb{O} \). Finally, we prove that \( \mathbb{R} \), \( \mathbb{C} \), \( \mathbb{H} \), and \( \mathbb{O} \) are the only algebraic real algebras with no nonzero divisor of zero satisfying the middle Moufang identity, or the right and left Moufang identities.
The metric dimension of a graph is the smallest number of vertices such that all vertices are uniquely determined by their distances to the chosen vertices. The corona product of graphs \( G \) and \( H \) is the graph \( G \odot H \) obtained by taking one copy of \( G \), called the center graph, \( |V(G)| \) copies of \( H \), called the outer graph, and making the \( j^{th} \) vertex of \( G \) adjacent to every vertex of the \( j^{th} \) copy of \( H \), where \( 1 \leqslant j \leqslant |V(G)| \). The Join graph \( G + H \) of two graphs \( G \) and \( H \) is the graph with vertex set \( V(G + H)=V(G) \cup V(H) \) and edge set \( E(G + H)=E(G) \cup E(H) \cup \{uv :u \in V(G),v \in V(H)\} \). In this paper, we determine the Metric dimension of Corona product and Join graph of zero divisor graphs of direct product of finite fields.
The secure edge dominating set of a graph \( G \) is an edge dominating set \( F \) with the property that for each edge \( e \in E-F \), there exists \( f \in F \) adjacent to \( e \) such that \( (F-\{f\})\cup \{e\} \) is an edge dominating set. In this paper, we obtained upper bounds for edge domination and secure edge domination number for Mycielski of a tree.
In this paper we contribute to the literature of computational chemistry by providing exact expressions for the detour index of joins of Hamilton-connected (\(HC\)) graphs. This improves upon existing results by loosening the requirement of a molecular graph being Hamilton-connected and only requirement certain subgraphs to be Hamilton-connected.
The geometrical properties of a plane determine the tilings that can be built on it. Because of the negative curvature of the hyperbolic plane, we may find several types of groups of symmetries in patterns built on such a surface, which implies the existence of an infinitude of possible tiling families. Using generating functions, we count the vertices of a uniform tiling from any fixed vertex. We count vertices for all families of valence \(5\) and for general vertices with valence \(6\), with even-sized faces. We also give some general results about the behavior of the vertices and edges of the tilings under consideration.
This study extends the concept of competition graphs to cubic fuzzy competition graphs by introducing additional variations including cubic fuzzy out-neighbourhoods, cubic fuzzy in-neighbourhoods, open neighbourhood cubic fuzzy graphs, closed neighbourhood cubic fuzzy graphs, cubic fuzzy (k) neighbourhood graphs and cubic fuzzy [k]-neighbourhood graphs. These variations provide further insights into the relationships and competition within the graph structure, each with its own defined characteristics and examples. These cubic fuzzy CMGs are further classified as cubic fuzzy k-competition graphs that show competition in the \(k\)th order between components, \(p\)-competition cubic fuzzy graphs that concentrate on competition in terms of membership degrees, and \(m\)-step cubic fuzzy competition graphs that analyze competition in terms of steps. Further, some related results about independent strong vertices and edges have been obtained for these cubic fuzzy competition graph classes. Finally, the proposed concept of cubic fuzzy competition graphs is supported by a numerical example. This example showcases how the framework of cubic fuzzy competition graphs can be practically applied to the predator-prey model to illustrate the representation and analysis of ambiguous information within the graph structures.
A graph \( X \) is \( k \)-spanning cyclable if for any subset \( S \) of \( k \) distinct vertices there is a 2-factor of \( X \) consisting of \( k \) cycles such that each vertex in \( S \) belongs to a distinct cycle. In this paper, we examine the \( k \)-spanning cyclability of 4-valent Cayley graphs on Abelian groups.
A path \(x_1, x_2, \dots, x_n\) in a connected graph \( G \) that has no edge \( x_i x_j \) \((j \geq i+3)\) is called a monophonic-triangular path or mt-path. A non-empty subset \( M \) of \( V(G) \) is a monophonic-triangular set or mt-set of \( G \) if every member in \( V(G) \) exists in a mt-path joining some pair of members in \( M \). The monophonic-triangular number or mt-number is the lowest cardinality of an mt-set of \( G \) and it is symbolized by \( mt(G) \). The general properties satisfied by mt-sets are discussed. Also, we establish \( mt \)-number boundaries and discover similar results for a few common graphs. Graphs \( G \) of order \( p \) with \( mt(G) = p \), \( p – 1 \), or \( p – 2 \) are characterized.
This note presents a counterexample to Propositions 7 and 8 in the paper [1], where the authors determine the values of \( V \) and \( W \). These values are crucial in determining the Hamming distance and MDS codes in the family of certain constacyclic codes over \(\mathbb{F}_{p^m}[u]/\langle u^3 \rangle\), which implies that the results found in [2] are incorrect. Furthermore, we provide corrections to the aforementioned results.
For a graph \( G \) and for non-negative integers \( p, q \) and \( r \), the triplet \( (p, q, r) \) is said to be an admissible triplet, if \( 3p + 4q + 6r = |E(G)| \). If \( G \) admits a decomposition into \( p \) cycles of length \( 3 \), \( q \) cycles of length \( 4 \), and \( r \) cycles of length \( 6 \) for every admissible triplet \( (p, q, r) \), then we say that \( G \) has a \( \{C_{3}^{p}, C_{4}^{q}, C_{6}^{r}\} \)-decomposition. In this paper, the necessary conditions for the existence of \( \{C_{3}^{p}, C_{4}^{q}, C_{6}^{r}\} \)-decomposition of \( K_{\ell, m, n}(\ell \leq m \leq n) \) are proved to be sufficient. This affirmatively answers the problem raised in \emph{Decomposing complete tripartite graphs into cycles of lengths \( 3 \) and \( 4 \), Discrete Math. 197/198 (1999), 123-135}. As a corollary, we deduce the main results of \emph{Decomposing complete tripartite graphs into cycles of lengths \( 3 \) and \( 4 \), Discrete Math., 197/198, 123-135 (1999)} and \emph{Decompositions of complete tripartite graphs into cycles of lengths \( 3 \) and \( 6 \), Austral. J. Combin., 73(1), 220-241 (2019)}.
The λ-fold complete symmetric directed graph of order v, denoted λKv*, is the directed graph on v vertices and λ directed edges in each direction between each pair of vertices. For a given directed graph D, the set of all v for which λKv* admits a D-decomposition is called the λ-fold spectrum of D. In this paper, we settle the λ-fold spectrum of each of the nine non-isomorphic orientations of a 6-cycle.
In this paper, we provide a correction regarding the structure of negacyclic codes of length \(8p^s\) over \(\mathcal{R} = \mathbb{F}_{p^m} + u \mathbb{F}_{p^m}\) when \(p^m \equiv 3 \pmod{8}\) as classified in [1]. Among other results, we determine the number of codewords and the dual of each negacyclic code.
The multiplicative sum Zagreb index is a modified version of the well-known Zagreb indices. The multiplicative sum Zagreb index of a graph \(G\) is the product of the sums of the degrees of pairs of adjacent vertices. The mathematical properties of the multiplicative sum Zagreb index of graphs with given graph parameters deserve further study, as they can be used to detect chemical compounds and study network structures in mathematical chemistry. Therefore, in this paper, the maximal and minimal values of the multiplicative sum Zagreb indices of graphs with a given clique number are presented. Furthermore, the corresponding extremal graphs are characterized.
Let \( G = (V, E) \) be a graph. A subset \( S \subseteq V \) of vertices is an \textit{efficient dominating set} if every vertex \( v \in V \) is adjacent to exactly one vertex in \( S \), where a vertex \( u \in S \) is considered to be adjacent to itself. Efficient domination is highly desirable in many real-world applications, and yet, in general, graphs are often not efficient. It is of value, therefore, to determine optimum ways in which inefficient graphs can be changed in order to make them efficient. It is well known, for example, that almost no \( m \times n \) grid graphs have efficient dominating sets. In this paper, we consider the minimum number of vertices that can be removed from an \( m \times n \) grid graph so that the remaining graph has an efficient dominating set.
Let \( G = (V, E) \) be any graph. If there exists an injection \( f : V \rightarrow \mathbb{Z} \), such that \( |f(u) – f(v)| \) is prime for every \( uv \in E \), then we say \( G \) is a prime distance graph (PDG). The problem of characterizing the family of all prime distance graphs (PDGs) with chromatic number 3 or 4 is challenging. In the fourth part of this series of articles, we determined which fans are PDGs and which wheels are PDGs. In addition, we showed: (1) a chain of \( n \) mutually isomorphic PDGs is a PDG, and (2) the Cartesian product of a PDG and a path is a PDG. In this part of the series, we improve (1) by showing that there exists a chain of \( n \) arbitrary PDGs which is a PDG. We also show that the following graphs are PDGs: (a) any graph with at most three cycles, (b) the one-point union of cycles, and (c) a family of graphs consisting of paths with common end vertices.
Let \(P_n\) and \(K_n\) respectively denote a path and complete graph on \(n\) vertices. By a \(\{pH_{1}, qH_{2}\}\)-decomposition of a graph \(G\), we mean a decomposition of \(G\) into \(p\) copies of \(H_{1}\) and \(q\) copies of \(H_{2}\) for any admissible pair of nonnegative integers \(p\) and \(q\), where \(H_{1}\) and \(H_{2}\) are subgraphs of \(G\). In this paper, we show that for any admissible pair of nonnegative integers \(p\) and \(q\), and positive integer \(n \geq 4\), there exists a \(\{pP_{4}, qS_{4}\}\)-decomposition of \(K_n\) if and only if \(3p+4q=\binom{n}{2}\), where \(S_4\) is a star with \(4\) edges.
Natural environment protection compensation refers to the legal system that protects the natural ecological environment, protects the natural environment and makes the beneficiaries of the natural environment get compensation by some means, so as to adjust the interests of the relevant subjects of natural ecological environment protection. This paper discusses the ecosystem service function and its type division of newly built areas in Ganjiang, and the emergy evaluation of ecosystem service function of newly built areas in Ganjiang, establishes the regular scheduling and joint optimal scheduling models of natural ecosystem service value single reservoir, intro-duces the corresponding model solving methods, and applies the ant colony algorithm to the optimal schedule is a lesson. According to the ant colony algorithm, the best way to study the region is to determine these algorithms. Combined with the kernel density analysis method, the spatial scope, potential corridors and key recovery points of ecological corridors are identified, and the optimization mode of natural ecological security pattern of Shule River is constructed. The experimental results show that the optimized ant colony algorithm proves that joint scheduling plays a more prominent role in ecological environment protection, mainly in ecological support and ecological regulation. At the same time, it verifies the applicability of ant colony algorithm in joint scheduling, and improves the average protection efficiency of natural ecology to 20.9%.
The semantic function of modern Chinese “negation + X” modal words based on communication technology and big data corpus has gained wide attention. As the basis of SOA architecture, Web services provide the key resources for worldwide information transfer and information sharing with their characteristics of loose coupling, platform independence, and data exchange without additional support from third-party hardware and software. However, along with the popularization and improvement of Web service technology, the number and types of Web services in the Internet are also increasing massively, and there are a large number of Web services with various functions, quality and granularity. Therefore, how to quickly and accurately discover Web services that satisfy users’ query requests from a large and complex set of services has become a critical problem to be solved in the current Web service discovery research. Based on the real corpus, this paper analyzes the similarities and differences in the semantic functions of modern Chinese “negation + X” modal words by combining lexicalization and grammatization, cognitive linguistics, systemic functional grammar and other related theories. The experimental results demonstrate that the model is designed for automatic annotation of semantic word classes, and the annotation algorithm based on the hidden horse model, combined with the Viterbi algorithm based on dynamic programming, achieves a correct rate of 94.3% in the closed test and 89.1% in the open test despite the small size of the training corpus and severe data sparsity, and the model fitting effect meets the dynamic expectations.
The relationship between competition state anxiety, motor motivation and coping styles of adolescent track and field athletes in China was investigated using interview and questionnaire research methods. The results showed that the mean scores of cognitive state anxiety and somatic state anxiety were lower in junior track and field athletes who had entered the echelon for a short period of time than in older athletes, and the opposite was true for state self-confidence; there were highly significant differences and significant differences in the identity regulation and introjection regulation dimensions of motor motivation; and there were significant differences in the focused problem-solving coping dimension of coping style. This paper proposes an algorithm for classifying athletic visual mirrors based on sequential model mining. This paper focuses on two issues – feature extraction and definition of semantic rules. In the feature extraction stage, the track and field video footage is automatically segmented into a series of identifiable sequences of athletic events, and then each type of behavioral event is identified using a mechanically learned algorithm. There were no significant differences between the three age groups in terms of race state anxiety, identity regulation and introjection regulation, and no significant differences in coping styles. There were no significant differences in the anxiety of competition status, motivation and coping styles among youth athletes of different sport levels. The results showed the effectiveness of the present algorithm for classifying track and field video cameras.
A graph labeling is an assignment of integers to the vertices or edges or both, which satisfies certain conditions. The domination cover pebbling number of a graph \( G \) is \( \psi(G) \), which is the minimum number of pebbles required such that any initial configuration of \( \psi(G) \) pebbles can be transformed through a number of pebbling moves so that the set of vertices with pebbles after the pebbling operation forms a dominating set of \( G \). In this paper, we explore the relationship between two graph parameters, namely graph labeling and domination cover pebbling.
In this paper, we study the \( A_\alpha \)-spectral radius of graphs in terms of given size \( m \) and minimum degree \( \delta \geq 2 \), and characterize the corresponding extremal graphs completely. Furthermore, we characterize extremal graphs having maximum \( A_\alpha \)-spectral radius among (minimally) \( 2 \)-edge-connected graphs with given size \( m \).
The metric dimension of a graph \(\Gamma = (V, E)\), denoted by \( \operatorname{dim}(\Gamma) \), is the least cardinality of a set of vertices in \(\Gamma\) such that each vertex in \(\Gamma\) is determined uniquely by its vector of distances to the vertices of the chosen set. The topological distance between an edge \(\varepsilon = yz \in E\) and a vertex \( k \in V \) is defined as \( d(\varepsilon, k) = \min\{d(z, k), d(y, k)\} \). A subset of vertices \( R_{\Gamma} \) in \( V \) is called an edge resolving set for \(\Gamma\) if for each pair of different edges \( e_{1} \) and \( e_{2} \) in \( E \), there is a vertex \( j \in R_{\Gamma} \) such that \( d(e_{1}, j) \neq d(e_{2}, j) \). An edge resolving set with minimum cardinality is called the edge metric basis for \(\Gamma\) and this cardinality is the edge metric dimension of \(\Gamma\), denoted by \( \operatorname{dim}_{E}(\Gamma) \). In this article, we show that the cardinality of the minimum edge resolving set is three or four for two classes of convex polytopes (\( S_{n} \) and \( T_{n} \)) that exist in the literature.
Language learning cannot be separated from the environment, and the environment for second language acquisition is becoming more and more perfect and ideal. It makes the traditional single and limited English learning environment gradually move towards a three-dimensional and diversified learning environment. On the premise of the great development of higher education in China, this study aims to conduct research and discussion on higher English teaching. In combination with other successful or well functioning higher English teaching reforms, it studies and discusses some problems faced in the construction and implementation of vocational English teaching application system in China, and finds solutions and methods. Therefore, this study has practical significance for the reform and development of higher English education in China. This manuscript is based on the design of the college English teaching system module of Web technology to realize the sharing of information resources. In addition, with the deepening understanding of the importance of English teaching in colleges and universities, improving English level and English teaching level is the goal of colleges and universities. English teachers urgently need to understand the factors that affect students’ English level in order to teach students in accordance with their aptitude and find the best teaching methods. The experimental results show that the system realizes the management, query and sharing functions of open level information, and has high security and interactivity. The development of the system conforms to the development trend of network information technology and promotes the informatization and standardization of college English teaching management.
At present, the social economy is entering the information age represented by computer, communication technology and network technology as the core, and the continuous development of modern information technology will certainly have a great impact on the teaching mode, content and methods of traditional accounting computerization. We aim to improve the existing higher vocational accounting teaching mode by building a multi-integrated teaching mode through federated learning based on 5G communication network as an environment for efficient information transfer. In addition, we develop a joint optimization strategy for priority-dependent task offloading, wireless bandwidth, and computational power in a distributed machine learning approach to ensure that more resources are allocated to users with higher priority while protecting user data privacy and reducing learning overhead. We have conducted extensive simulation experiments for both environments, and these simulation results demonstrate the effectiveness of our proposed solutions for different problems from different perspectives.
This study explores the integration of “Internet+” into university education to enhance students’ learning, innovation, and entrepreneurship (I&E) skills. By updating educational concepts and methods, we aim to establish a comprehensive I&E framework that includes mindset development, knowledge acquisition, skill enhancement, and team support. Practical training and network learning communities are emphasized to provide global platforms for skill improvement and project incubation. Through a case study, we analyze the development and effectiveness of an I&E education platform, highlighting the importance of targeted demand surveys and data analysis. Our findings demonstrate the significance of aligning talent training with industry needs, fostering creativity, and promoting entrepreneurial success through collaborative school-enterprise initiatives.
The concept of graph energy, first introduced in 1978, has been a focal point of extensive research within the field of graph theory, leading to the publication of numerous articles. Graph energy, originally associated with the eigenvalues of the adjacency matrix of a graph, has since been extended to various other matrices. These include the maximum degree matrix, Randić matrix, sum-connectivity matrix, and the first and second Zagreb matrices, among others. In this paper, we focus on calculating the energy of several such matrices for the join graph of complete graphs, denoted as \( J_{m}(K_{n}) \). Specifically, we compute the energies for the maximum degree matrix, Randić matrix, sum-connectivity matrix, first Zagreb matrix, second Zagreb matrix, reverse first Zagreb matrix, and reverse second Zagreb matrix for \( J_{m}(K_{n}) \). Our results provide new insights into the structural properties of the join graph and contribute to the broader understanding of the mathematical characteristics of graph energy for different matrix representations. This work extends the scope of graph energy research by considering these alternative matrix forms, offering a deeper exploration into the algebraic and spectral properties of graph energy in the context of join graphs.
Tunnels are essential infrastructure elements, and it is critical to maintain their stability for both operation and safety. Using engineering techniques, this study examines the correlation between rock mass motorized characteristics and tunnel surrounding rock stability. This study utilizes the multi-sensor monitoring data of the surrounding rock mechanical characteristics and tunnel support structure collected during tunnel boring machine construction as its research object. The integrated cuckoo search optimized Upgraded dynamic convolutional neural network (ICSO-UDCNN) has been utilized for predicting the tunnel parameters. In general, the surrounding rock’s hardness correlates with its level, which in turn determines how quickly tunnels are being excavated. There is a stronger correlation of 98\% between the field penetration index (FPI) variables of the rock’s characteristic slope along the conditions surrounding the tunnel. The most significant factor influencing its deformation is the surrounding rock’s mechanical characteristics. For engineers and other decision-makers engaged in tunnel design, building, and maintenance, the study’s findings add a greater understanding of the variables affecting tunnel stability. This research provides an establishment for enhancing security protocols, lowering hazards related to tunneling operation, and optimizing tunnel engineering techniques by quantitatively evaluating the influence of rock mass mechanical factors on solidity.
Let \( G \) be a graph and let \( 0 \leq p, q \) and \( p + q \leq 1 \). Suppose that each vertex of \( G \) gets a weight of \( 1 \) with probability \( p \), \( \frac{1}{2} \) with probability \( q \), and \( 0 \) with probability \( 1-p-q \), and vertex weight probabilities are independent. The \textit{fractional vertex cover reliability} of \( G \), denoted by \( \operatorname{FRel}(G;p,q) \), is the probability that the sum of weights at the end-vertices of every edge in \( G \) is at least \( 1 \). In this article, we first provide various computational formulas for \( \operatorname{FRel}(G;p,q) \) considering general graphs, basic graphs, and graph operations. Secondly, we determine the graphs which maximize \( \operatorname{FRel}(G;p,q) \) for all values of \( p \) and \( q \) in the classes of trees, connected unicyclic and bicyclic graphs with fixed order, and determine the graphs which minimize it in the classes of trees and connected unicyclic graphs with fixed order. Our results on optimal graphs extend some known results in the literature about independent sets, and the tools we developed in this paper have the potential to solve the optimality problem in other classes of graphs as well.
Let \( G \) be an undirected graph. A \textit{vertex tree cover} of \( G \) is a collection of trees such that every vertex of \( G \) is incident with at least one tree. Similarly, an edge tree cover is a collection of trees such that every edge of \( G \) is contained in at least one tree. The tree cover number is defined as the minimum number of trees required in such a cover. In this paper, we demonstrate that when considering specific types of tree covers, only vertex permutations act as linear operators that preserve the tree cover number of \( G \).
To fulfil the requirements of task scheduling for processing massive amounts of graph data in cloud computing environments, this thesis offers the LGPPSO method, which is based on Particle Swarm Optimisation. The LGPPSO algorithm considers the task’s overall structure when initialising numerous particles in order to broaden the search range and raise the likelihood of finding an approximation optimal solution. We evaluated it in large-scale simulation trials with 100 performance-heterogeneous virtual machines (VMs) using both randomly generated and real application graphs, and evaluated its effectiveness against the CCSH and HEFT algorithms. The experimental findings demonstrate that, in both randomly generated graphs and real graph structure applications, significantly reducing the scheduling duration of large-scale graph data is the LGPPSO algorithm. For randomly generated 200 and 400 node tasks, respectively, the scheduling length is shortened by approximately 8.3% and 9.7% on average when compared to the HEFT algorithm. The LGPPSO technique minimises the scheduling length for actual graphical structure applications by an average of 14.6% and 16.9% for the Gaussian 100 and 200 node examples, respectively.
The application of graph theory is instrumental in the construction of a practical teaching mode structure for an ideological and political theory (IPE) course rooted in fuzzy system theory. In recent years, IPE education has received more and more attention. System theory can well construct the IPE teaching mode, so we need to have a good grasp of system theory. This paper starts with the significance of system theory in the practical teaching of IPE, finds the IPE curriculum that conforms to system theory, and constructs the basic mode of practical teaching of IPE on this basis. Using the idea of a multi-level fuzzy comprehensive evaluation to quantify teachers’ teaching quality, this paper establishes an algorithm model to quantitatively measure teachers’ teaching quality fuzzy.
The new curriculum standards have put forward new requirements for high school English vocabulary teaching, and the English vocabulary eco-classroom under the guidance of ecological linguistics theory can precisely make up for the shortcomings in the traditional vocabulary classroom and meet the challenges of the times. However, most of the existing researches on ecological classroom are combined with macro English subjects, and few of them are about English vocabulary teaching. This study takes the principles of ecological linguistics as the theoretical basis to support the conceptual construction and morphological reliance of the vocabulary ecological classroom, supplemented by modal theory as the process orientation of the four major stages in the teaching process, constructs a new vocabulary ecological classroom model based on Markov chain model, and applies the ecological classroom model to high school English vocabulary teaching to verify its teaching effects. The experimental results show that the Markov chain-based high school English vocabulary teaching under the “ecological linguistics” model can help students’ interest in vocabulary learning and promote their vocabulary learning level, accounting for 15% improvement in the learning effect.
With the development of social economy, urban planning has been paid more and more attention. At present, China is implementing low-carbon environmental planning in stages, stages and stages with the development goal of building two types of society, namely, national master plan and strategic development. This paper analyzes low-carbon environmental planning based on blockchain, then expounds the specific content of whole process project management, and completes the design of green building index system based on low-carbon environmental planning. At the same time, the design method of new dynamic structure and index system of green building is expounded. The comparative analysis shows that the material loss cost of the new environmental protection building engineering and index system is low, and the material utilization rate is high, accounting for more than 95%.
The question on how to colour a graph \( G \) when the number of available colours to colour \( G \) is less than that of the chromatic number \(\chi(G)\), such that the resulting colouring gives a minimum number of edges whose end vertices have the same colour, has been a study of great interest. Such an edge whose end vertices receive the same colour is called a bad edge. In this paper, we use the concept of \(\delta^{(k)}\)-colouring, where \( 1 \leq k \leq \chi(G)-1 \), which is a near proper colouring that permits a single colour class to have adjacency between the vertices in it and restricts every other colour class to be an independent set, to find the minimum number of bad edges obtained from the same for some wheel-related graphs. The minimum number of bad edges obtained from \(\delta^{(k)}\)-colouring of any graph \( G \) is denoted by \( b_k(G) \).
Let \( G \) be a simple finite graph. A \( k \)-coloring of \( G \) is a partition \(\pi = \{ S_1, \cdots, S_k \}\) of \( V(G) \) so that each \( S_i \) is an independent set and any vertex in \( S_i \) takes color \( i \). A \( k \)-coloring \(\pi = \{ S_1, \cdots, S_k \}\) of \( V(G) \) is a neighbor locating coloring if for any two vertices \( u, v \in S_i \), there is a color class \( S_j \) for which one of them has a neighbor in \( S_j \) and the other does not. The minimum \( k \) with this property is said to be the neighbor locating chromatic number of \( G \), denoted by \(\chi_{NL}(G)\). We initiate the study of the neighbor locating coloring of graphs resulting from three types of product of two graphs. We investigate the neighbor locating chromatic number of Cartesian, lexicographic, and corona products of two graphs. Finally, we untangle the neighbor locating chromatic number of any of the aforementioned three products of cycles, paths, and complete graphs.
The significance of advancing socialist S&T legal system building with Chinese features is examined in this paper, along with the extensive effects it has on S&T innovation and economic growth. Against the backdrop of an increasingly competitive global scientific and technology landscape, China’s ability to innovate and achieve sustainable growth hinges on the establishment of an ideal science and technology legislative framework. This essay first examines the primary obstacles to China’s development of a science, technology, and innovation (S&T) legal system, including inadequate protection for intellectual property rights, a flawed process for transforming scientific and technological advancements, and an insufficient system for encouraging enterprise innovation. Then, this research presents a quantitative analysis model to optimize the path of science and technology legal building by applying the improved particle swarm optimization method (PSO). The model takes into account a wide range of variables, including the degree of intellectual property protection, the strength of legal backing, the pace at which scientific and technological advancements are transformed, etc. Through the analysis of simulation data, the model also confirms the promotion effect of the legal system construction on the quantity of patent applications, the success rate of innovation projects, the enterprise R&D expenditure, and the expansion of the local economy. The study’s findings demonstrate that bolstering the science and technology legal system can effectively encourage businesses to boost R&D investment and foster regional economic development in addition to greatly raising the quantity of patent applications and the success rate of innovation projects. The rigorous intellectual property protection laws and ideal legal framework for the conversion of accomplishments greatly boost the regional innovation vitality and economic efficiency, particularly in the case study of Zhongguancun in Beijing and East China. Moreover, adaptive weighting is used to enhance the PSO algorithm and optimize the development of science and technology legal system’s comprehensive performance index, thereby confirming the model’s viability and efficacy. The study’s findings offer theoretical justification and helpful advice for China’s development of a science and technology legal framework, which is crucial for fostering innovation in these fields and boosting the country’s competitiveness.
At present, there are few systematic researches on macro-scale heterogeneous modeling and numerical simulation of dynamic mechanical properties of 3-D braided composites. In this paper, the parametric virtual simulation model of 3D five-directional braided composites is realized in the way of “point-line-solid” based on the integrated design idea of process-structure-performance. And the impact compression numerical simulation of the material is carried out by using multi-scale analysis method. The effects of strain rate and braiding angle on transverse impact compression properties and fracture characteristics of composites is studies and verified by comparing the test results with the numerical simulation results systematically. The dynamic failure mechanism of the matrix and fiber bundles during the impact compression process is revealed. The results show that the macro-scale heterogeneous simulation model of 3D five-directional braided composites established is effective, and the numerical simulation results agree well with the test results. The matrix fracture and shear failure of fiber bundles are presented simultaneously under transverse impact compression. The failure of fiber bundles and matrix mainly concentrates on two main fracture shear planes. And the included angle between the fracture shear planes and the vertical direction is consistent with the corresponding internal braiding angle of specimens.
The paper extensively examined the intricate components underpinning innovation ability, culminating in the construction of a linear spatial model delineating innovation and entrepreneurship prowess. This paper analyzed the components of the connotation of innovation ability, then constructs a linear spatial model of innovation and entrepreneurship ability, proposes a multi-objective function model of the utilization efficiency and allocation efficiency of education resources, and uses the grey correlation algorithm The experimental simulation and model solution are carried out. The simulation results show that, through the optimization, the utilization efficiency and allocation efficiency of the educational resources for innovation and entrepreneurship for all are increased by 18.72% and 20.98% respectively, and tend to be in equilibrium, which can achieve the optimization of educational resources allocation. Among all people, the correlation value with ideal entrepreneurship is 0.3177, achieving the most excellent innovation and entrepreneurship education.
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