Let \(\mathcal{F}\) be a family of graphs, and \(H\) a “host” graph. A spanning subgraph \(G\) of \(H\) is called \(\mathcal{F}\)- saturated in \(H\) if \(G\) contains no member of \(\mathcal{F}\) as a subgraph, but \(G+e\) contains a member of \(\mathcal{F}\) for any edge \(e\in E(H) – E(G)\). We let \(Sat(H,\mathcal{F})\) be the minimum number of edges in any graph \(G\) which is \(\mathcal{F}\)-saturated in \(H\), where \(Sat(H,\mathcal{F}) = |E(H)|\) if \(H\) contains no member of \(\mathcal{F}\) as a subgraph. Let \(P_{m}^{r}\) be the \(r\)-dimensional grid, with entries in each coordinate taken from \(\{1,2,\cdots , m\}\), and \(K_{t}\) the complete graph on \(t\) vertices. Also let \(S(F)\) be the family of all subdivisions of a graph \(F\). There has been substantial previous work on extremal questions involving subdivisions of graphs, involving both \(Sat(K_{n},S(F))\) and the Turan function \(ex(K_{n},S(F))\), for \(F = K_{t}\) or \(F\) a complete bipartite graph. In this paper we study \(Sat(H, S(F))\) for the host graph \(H = P_{m}^{r}\), and \(F = K_{4}\), motivated by previous work on \(Sat(K_{n}, S(K_{t}))\). Our main results are the following; 1) If at least one of \(m\) or \(n\) is odd with \(m\geq 5\) and \(n\geq 5\), then \(Sat(P_{m}\times P_{n}, S(K_{4})) = mn + 1.\) 2) For \(m\) even and \(m\geq 4\), we have \(m^{3} + 1 \le Sat(P_{m}^{3}, S(K_{4}))\le m^{3} + 2.\) 3) For \( r\geq 3\) with \(m\) even and \(m\geq 4\), we have \(Sat(P_{m}^{r}, S(K_{4})) \le m^{r} + 2^{r-1} – 2\).

An undirected graph is said to be cordial if there is a friendly (0,1)-labeling of the vertices that induces a friendly (0,1)-labeling of the edges. An undirected graph \(G\) is said to be \((2,3)\)-orientable if there exists a friendly (0,1)-labeling of the vertices of \(G\) such that about one-third of the edges are incident to vertices labeled the same. That is, there is some digraph that is an orientation of \(G\) that is \((2,3)\)-cordial. Examples of the smallest noncordial/non-\((2,3)\)-orientable graphs are given, and upper bounds on the possible number of edges in a cordial/\((2,3)\)-orientable graph are presented. It is also shown that if \(T\) is a linear operator on the set of all undirected graphs on \(n\) vertices that strongly preserves the set of cordial graphs or the set of \((2,3)\)-orientable graphs, then \(T\) is a vertex permutation.

With the social progress and technological development, China’s criminal activities gradually show the characteristics of specialization, networking, and hotspotting, which leads to the phenomenon of high incidence but low detection rate, and the prediction of the criminal phenomenon is particularly important. In this paper, we construct a graph self-encoder, and derive the formula of the GAE loss function from the corresponding reconstructed neighbor matrix and node feature loss function of GAE. The spatial channel attention mechanism is introduced to improve the performance of the model, and the time window dimension is mapped to the perceptual self-attention module, and the objective function is constructed by generating a collection of crime matrices for future time windows. A multi-raster layer analysis model is added to optimize the model, generate a risk map of criminal activities, quantify the risk value of each element, and form a spatio-temporal prediction effect. Comparison experiments are used to analyze the optimization effect of the model, and the absolute error of the optimized model is no more than 0.05 for four types of cases. The prediction results of the cases of property invasion in different time periods show that the number of cases occurring in the early hours of the morning is 508, and the average PEI index is 0.19, which is smaller compared with other time periods.

With the rapid urbanization and expansion of subway rail transit, the subway has become an essential mode of public transportation. This study explores the impact of subway car color design on passengers’ psychological responses. Utilizing computer vision technology and a pruning algorithm, a target detection model for passenger expression recognition was developed, serving as an intuitive measure of psychological reactions. An optimized expression feature extraction network was constructed for facial expression recognition, while a multidimensional data analysis model, based on data mining, provided comprehensive insights. The study reveals that green, red, and yellow lighting evoke positive psychological responses, whereas blue and purple induce calmer or more somber reactions. These findings offer valuable guidance for urban subway carriage color lighting design, enhancing passenger experience.

As economic globalization progresses, air transport has become increasingly vital to economic development due to its speed and convenience. This study examines the driving forces of airside economic construction across four levels: primary, secondary, derivative, and permanent influences. It explores the dynamic interplay between the aviation industry and airside economic construction. Using the entropy weight method to optimize the grey situation decision-making theory, the paper investigates the development strategies for Henan Province’s airside economy. Results indicate that the H2 area should be prioritized as the key construction zone, achieving the highest effect measurement score of 0.9789. Furthermore, focusing on the development of the tertiary industry or the joint advancement of secondary and tertiary industries in the H2 area yields the most significant economic impact, with effect measurement scores of 0.755 and 0.749, respectively.

This paper explores the integration of blockchain technology into the teaching quality evaluation system of universities. A practical teaching quality evaluation index system for applied technology universities is developed, ensuring data authenticity through blockchain’s de-trusting mechanism. To enhance data storage efficiency, the PBFT consensus algorithm is improved and incorporated into a technical architecture adopting an “off-chain storage + on-chain sharing” model. The algorithm scoring formula and improved PBFT consensus algorithm are analyzed to demonstrate their effectiveness. Practical applications in applied technology universities highlight the benefits of blockchain in higher education evaluation. The CBFT-based consensus algorithm achieves average CPU utilization of 13.4% compared to 18.5% in traditional algorithms, while ensuring data transparency and tamper-proofing. Additionally, the algorithm improves transaction throughput and reduces resource consumption, enabling efficient operation of the teaching evaluation system in applied sciences universities.

Translation as a cross-cultural information exchange and exchange activity has the nature of dissemination. Combining communication and translation helps make translation an open, dynamic, and comprehensive discipline. Translators play the role of gatekeepers in communication studies. The choice of a translator is affected by any change in the translator himself, such as his personal preference, motivation, life experience, aesthetic orientation, psychological factors and values, which can call for different translations to be produced. The translation of classics is not like the translation of ordinary works. It puts forward higher requirements for the translator. The beauty and subtlety of its words and characters require the translator to have a profound knowledge of the target language; its connotation and thought are broad and profound, and the translator needs to understand the source language. Transparency of this understanding. And such a master is really rare, and it is difficult to cultivate, so excellent translation works of classics are not common. In addition, translations are becoming more and more diverse, and there is inevitably a mix of people and irregularities in the intermediate translations. This paper explores the translation of classics that combines machine learning technology with the perspective of communication, and proposes an efficient translation model. The experimental results show that the model can effectively improve translation efficiency and accuracy.

We consider the generating function for increasingly labelled trees. By generalizing the proof through symbolic method, we are able to study various statistics regarding binary increasing trees with respect to height restrictions. We then apply our approach to special colorings of increasing trees in order to obtain their generating functions and, from there, derive the counting sequence for \((ak+a)\)-colored recursive trees. We also present some interesting bijections between colored and non-colored increasing trees.

This paper aims to enhance the moral and vocational qualities of college students by integrating moral education elements into career planning education. The BOPPPS teaching model is constructed, comprising six modules: introduction, objectives, pre-test, participatory learning, post-test, and summary, to effectively stimulate students’ interest and initiative. Moral education elements are integrated into career planning education through an intelligent teaching platform, incorporation into teaching processes, and the use of the second classroom to promote in-class and out-of-class linkages. Additionally, a fuzzy classroom teaching evaluation system is developed to assess the effectiveness of career planning education. The results indicate high reliability and validity of the evaluation system, with an alpha coefficient exceeding 0.8, a KMO value of 0.938, and a Bartlett’s test P-value of 0.000. Students’ positive classroom mood improved significantly from 35.79% to 68.42%, alongside an enhanced evaluation of classroom learning. The findings demonstrate the practical value of this approach in advancing education reform.