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.