Let\(G\) be an undirected graph. A tree partition of\(G\) is a set of trees whose edge sets are disjoint and whose union is the edge set of\(G\). The minimum cardinality of such a tree partition is called the tree partition number of\(G\). We show that for various types of trees allowed in the tree partition, that the only linear operators that preserve the tree partition number are vertex permutations.

The Mostar index \( \text{MoI} \) of a finite and connected graph \( G \) is a measure of asymmetry, focusing on the edge-based structure of the graph. For an edge \( xy \) in \( G \), let \( \gamma_{xy} \) and \( \gamma_{yx} \) denote the cardinalities of the sets of vertices closer to \( x \) and \( y \) respectively, then the Mostar index is defined as: \( \text{MoI}(G) = \sum_{xy \in E(G)} |\gamma_{xy} – \gamma_{yx}| \) where the summation is taken over all edges \( xy \in G \). This edge-wise difference reflects how asymmetrically the graph is structured around each edge and summing these differences across all edges yields the Mostar index for the graph. In this article, we compute the \( \text{MoI} \) for certain classes of bicyclic graphs that are of particular interest due to their moderately complex structure, lying between acyclic and polycyclic graphs. We classify bicyclic graphs into three distinct types, namely \( \mathcal{B}^{1}(m,n),\; \mathcal{B}^{2}(l,m,n) \) and \( \mathcal{B}^{1}(l,m) \), based on their cycle arrangements and then provide explicit formulas for calculating the exact value of the Mostar index.

Optimizing regional economic resources is a crucial aspect of the Belt and Road initiative. This paper develops a multi-objective optimization model to objectively evaluate the development level of regional economic resource optimization in Belt and Road countries and to identify the key influencing factors. The model maximizes regional economic and social benefits under constraints of resource availability, output capacity, and coordinated regional development, and it incorporates a synergy measure to ensure robust progress. Our findings show that the regional economic benefits index increased from 0.264 in 2017 to 0.575 in 2023 (a growth rate of 117.8%), while social benefits grew by 14.29%. Additionally, panel regression analysis reveals that merchandise trade, foreign direct investment, road traffic mortality, and industrial development all have significant negative impacts on the optimization of economic resources, at the 1% significance level.

Alzheimer’s disease (AD) is a progressive neurodegenerative condition that affects the elderly population. The early detection and diagnosis of AD is critical for achieving effective treatment, as it can greatly improve the patient experience. AD can be viewed through imaging techniques like MRI, PET, and SPECT, providing valuable information about structural and functional changes. These findings are important in understanding this area. However, each imaging modality offers a different perspective. This information can be better collected from several of the other modalities as well as from some others to improve accuracy and reliability in AD detection. By combining information from different imaging modalities, such as MRI, PET, DTI, and fMRI, automated multimodal medical image frameworks aim to create a fused representation that preserves the relevant features from each modality. Convolutional neural networks (CNNs) and generative adversarial networks (GANs), among other deep learning techniques, have been prevalent in these frameworks for learning discriminative and informative features from multi-modal data. In this paper, The Alzheimer’s Disease Neuroimaging Initiative (ADNI) is used for experimental analysis. The proposed work gives 98.94% of accuracy and 1.06% of error which is greater than the existing approaches.

The power of the public key cryptosystem based on Paley graphs is due to several mathematical problems namely quadratic residuosity, local equivalence, and identification of the graphs induced by a sequence of local complementations of the Paley graphs. The classification in terms of degree of these induced graphs can be useful in the cryptanalysis part of the proposed public-key cryptosystem based on these algebraic graphs. This work aims to give the exact value of the minimum and maximum degree by local complementation, then the possible classifications in terms of degree to the graphs induced by a sequence of local complementations of Paley graphs of degree p less than or equal to 13 and some information about the equivalence problem.

Given a graph \(G \), a set is \(\Delta \) convex if there is no vertex \(u\in V(G)\setminus S \) that forms a triangle with two vertices of \(S \). The \(\Delta \)-convex hull of \(S \) is the minimum \(\Delta \)-convex set containing \(S \). This article is an attempt to discuss the Carath\’eodory number and exchange number on various graph families and standard graph products namely Cartesian, strong and lexicographic products of graphs.

Directed strongly regular graphs were introduced by Duval in 1998 as one of the possible generalization of classical strongly regular graphs to the directed case. Duval also provided several construction methods for directed strongly regular graphs. In this paper, an infinite family of directed strongly regular graphs is constructed, as generalized Cayley graphs.

Green construction is becoming a mainstream model of the transformation and upgrading of the construction industry, which has the advantages of energy saving, environmental protection and ecology, which can effectively reduce energy deficiency and improve environmental quality, which is the need for high quality sustainable development. This study is based on BIM software and the intelligent construction technology to propose the green architectural design party case. Building energy-saving efficiency evaluation system, using fuzzy Borda method and the CRITIC method of evaluation, the objective of the index, and the example of a community, the use of the object meta-effect model. The evaluation scores of the energy saving efficiency of the building of green energy saving and renovation are in the 90.11-99.28 points, and the high energy demand in the process of running the use of the building is excellent in the heating, refrigeration and other aspects of the building. This paper shows that the goal of the green transformation project is basic, which is effective and the efficiency of energy efficiency is generated. This study can provide guidance for the work of the green building energy saving and renovation work, and further promote the energy saving and transformation of China.

The supply chain applies large number according to the technology, can reduce the cost of each link, optimize the resource allocation, increase the enterprise benefit. In this paper, the supply chain cost control program based on large number according to the previous forecast, the control of the event and the analysis of the three levels of the analysis of the supply chain. The combination time series model and the multivariate regression model, the joint CPFR concept, the establishment of the CPFR sales combination demand prediction model, the design form according to the sales prediction system, the resource optimization plan of the supply chain inventory in real time. Analyze the prediction effect of the combination prediction model, predict the product sales in the week, calculate the product safety inventory and the remaining inventory. The analysis is based on the cost control effect of the enterprise supply chain according to the sales forecast. The cost of purchasing the supply chain costs less than the operating income, which fell to 0.5107in 2023. The gross margin of gross profit was 0.53666 in 2023, which was controlled by the gross margin, and the gross profit was improved. It is said that the enterprise is using large number according to the technology to the supply chain resource optimization in the supply chain cost control, the cost control effect is better.

The research of modular and personalized balance strategy in assembly building design can improve the efficiency of construction and meet the demand of design diversification. Based on bim technology, an assembly building modular design method is proposed to determine the required space module, to determine the required space module, to strengthen the module structure, to set up the layout of the building, to formulate the modular panel and the assembly frame platform, and through the revit implementation of the three-dimensional visual design of modularity and personalization. The design of the 9 building of this article, in the collaborative function, spatial adaptability and the design diversity score in turn for 10th 10 “10” 9 points (full score 10). This article is designed to meet the demand of the building in daylighting and ventilation, the average daylighting coefficient is 6.440%, and the minimum value of the floor area of the building room is 18.25cent. Modular and personalized assembly frame structures have a better seismic resistance, and their limit cumulative energy consumption is 2.38 times the traditional way. Experts have the highest social benefit satisfaction in this article strategy, and the satisfaction score is 92.05.