Let \(\Gamma_{G}\) be the orbit graph of \(G\), with non-central orbits in the subset of order two commuting elements in \(G\), and the vertices of \(\Gamma_{G}\) connected if they are conjugate. The main objective of this study is to compute several topological indices for the orbit graph of a dihedral group, including the Wiener index, the Zagreb index, the Schultz index, and others. We also develop a relationship between the Wiener index and the other indices for the dihedral group’s orbit graph. Furthermore, their polynomial has been computed as well.

\(Y_k\)-tree is defined as \((v_1, v_2,\ldots, v_{k-1};\, v_{k-2} v_k)\) by taking their vertices as \((v_1,\,v_2,\ldots,\,v_k)\) and edges as \(\{(v_1v_2, v_2v_3,\ldots, v_{k-2}v_{k-1})\cup (v_{k-2}v_k)\}\). It is also represented as \((P_ {k-1} +e)\). One can obtain the necessary condition as \(mn(m-1)(n-1)\equiv 0 \pmod {2(k-1)}\), for \(k \geq 5\) to establish a \(Y_k\)-tree decomposition in \(K_m \times K_n\). Here the tensor product is denoted by \(\times\). In this manuscript, it is shown that a \(Y_5\)-tree (gregarious \(Y_5\)-tree) decomposition exists in \(K_m \times K_n\), if and only if, \(mn(m-1)(n-1)\equiv 0 \pmod8\).

For graphs \(F\) and \(H\), the proper Ramsey Number \(PR(F,H)\) is the smallest integer \(n\) so that any \(\chi'(H)\)-edge-coloring on \(K_n\) contains either a monochrome \(F\) or a properly colored \(H\). We determine the proper Ramsey number of \(K_3\) against \(C_3\) and \(C_5\).

We have constructed Block structured Hadamard matrices in which odd number of blocks are used in a row (column). These matrices are different than those introduced by Agaian. Generalised forms of arrays developed by Goethals-Seidel, Wallis-Whiteman and Seberry-Balonin heve been employed. Such types of matrices are applicable in the constructions of nested group divisible designs.

The primary challenge in credit analysis revolves around uncovering the correlation between repayment terms and yield to maturity, constituting the interest rate term structure-an essential model for corporate credit term evaluation. Presently, interest rate term structures are predominantly examined through economic theoretical models and quantitative models. However, predicting treasury bond yields remains a challenging task for both approaches. Leveraging the clustering analysis algorithm theory and the attributes of an insurance company’s customer database, this paper enhances the K-means clustering algorithm, specifically addressing the selection of initial cluster centers in extensive sample environments. Utilizing the robust data fitting and analytical capabilities of the Gaussian process mixture model, the study applies this methodology to model and forecast Treasury yields. Additionally, the research incorporates customer credit data from a property insurance company to investigate the application of clustering algorithms in the analysis of insurance customer credit.

In this paper, we propose a method for effectively evaluating the quality of business English teaching in colleges and universities. The approach is based on a multimodal neural network model integrated with grey correlation analysis. By employing the optimal data clustering criterion, we identify teaching quality evaluation indices. Subsequently, we establish a teaching quality evaluation index system using a genetic algorithm (GA) optimized Radial Basis Function (RBF) neural network. Grey correlation analysis is then applied to assess the quality of business English teaching by considering the relationship between the correlation degree and the evaluation level. The results indicate a correlation degree exceeding 0.90, signifying excellent teaching quality. The reliability of the selected evaluation indicators, assessed through retesting, surpasses 0.700, validating the evaluation results.

A crucial component of kindergarten instruction, collective teaching activities are a good way to educate young children on their overall development. The language field is one of the subjects taught in kindergarten, and it has to do with how kids learn to read, write, and speak. In order to improve teachers’ comprehension of children’s emotional reactions and language, this paper combines quantitative and qualitative methods to observe and analyze the quality of current language collective teaching activities in kindergartens. It also suggests knowledge logic and psychological logic for grasping the content of language collective teaching in kindergartens. To improve the quality of language teaching in kindergartens, it is crucial to adopt a variety of teaching strategies and organizational techniques, provide the proper tools and materials for language learning, pay attention to the key experiences of children learning the language, and enhance learning quality.

In the new era characterized by the modernization of national governance, fair competition is the inherent requirement of building a modern market system. However, the abuse of administrative power by administrative organs to excessively interfere in free-market competition is widespread, seriously damaging the market competition order in China. To avoid the unreasonable intervention of administrative organs in the market economy, restrain the administrative acts of administrative organs, and form a highly “competitive” market environment, the fair competition review system came into being. With the rapid development of blockchain technology, new ideas are provided for the research of fair-trade protocols. Aiming at the system performance bottleneck and high-cost problems caused by centralized processing in traditional fair transaction schemes based on trusted third parties, a fair transaction scheme based on fuzzy signature is proposed. In the proposed scheme, the signature model uses concurrent signature, and both parties hold their own key numbers, which are released through blockchain transactions to bind their signatures. In the whole process, both parties can complete the contract signing without the assistance of a centralized third party. Based on analyzing the security of the proposed scheme, the performance of the proposed scheme is further compared with other similar schemes of the same kind, which shows that the proposed scheme has higher computational efficiency.

Selecting the user comment information of short videos with top 2 likes in the top 50 topics about public cultural services in Shake App as the research object, and facilitating video platforms to identify the high and low quality of the videos and make reasonable promotion arrangements by predicting the short-term playback volume of pop-up videos and analyzing the influencing factors, which is conducive to improving the platform’s pop-up video services and economic benefits. The data related to B station videos are captured, and feature selection and different algorithms are combined to construct random forest model, XG Boost model and LSTM model to predict the playback volume of the pop-up videos, and compare and analyze the effects of different feature combinations on the experimental results. The results show that the prediction accuracy of the random forest model is higher than that of the XG Boost model and the LSTM model, and the features of the pop-up video itself have the greatest influence on the playback volume, while the features of the video markup text have a smaller degree of influence on the playback volume.

The rapid economic progress and widespread use of sophisticated technology elevate the output value per kWh of electricity consumed, underscoring the paramount importance of maintaining an uninterrupted and dependable power supply to avoid substantial economic losses for consumers and society. Investigating the reliability of urban distribution systems emerges not only as a pivotal factor in enhancing power supply quality but also as a cornerstone of electric power modernization, significantly impacting production, technology, and management within the industry while bolstering its economic and social benefits. This study adopts a multifaceted approach: firstly, establishing a methodology for grid-side storage capacity distribution to mitigate substation load factors and implement peak shaving, thereby minimizing load discrepancies. Secondly, it develops a mathematical model encompassing diverse user distributions, employing analytical techniques to derive reliability indices and optimal segment numbers tailored to different user distributions. The research proposes segment optimization based on user distributions, considering both economic viability and reliability, showcasing an interdisciplinary amalgamation of combinatorial principles and scientific computing methodologies. This approach aims to optimize segment distribution, enhancing the reliability and economic feasibility of urban distribution networks through advanced mathematical and computational techniques.