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.