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.

The combination of thermal power units’ stability and energy storage systems’ rapid response time enhances power system frequency control. However, high costs and battery life impacts from charging/discharging strategies limit energy storage adoption. This study proposes an adaptive weight-based particle swarm optimization algorithm (APSO) to optimize energy storage control for joint thermal-storage frequency modulation (FM). By analyzing the coupling between state of charge (SOC) and charging/discharging power, the study implements “shallow charging and discharging” with dynamic SOC constraints. The improved PSO algorithm integrates adaptive weighting to overcome local optimal convergence, enhancing global search capabilities and particle migration. Simulation results, based on real-world power plant data, show improved FM accuracy, faster regulation, and reduced energy storage system loss, significantly boosting economic efficiency.

With the increasing penetration of distributed intermittent energy into distribution networks, the self-healing problem of distribution networks faces significant challenges. The load level and demand response must be considered as critical factors affecting fault recovery. This paper proposes a fault recovery strategy that combines islanding division and network reconstruction. First, a distribution network model with a distributed energy storage system is established. To optimize the use of distributed energy resources, controllable loads that can respond to demand are prioritized, and high-priority loads are included in the islanded network after a fault. Based on the islanding division results, the remaining non-faulty power loss areas are restored through main network reconstruction. The improved whale optimization algorithm is employed to solve the problem. Simulation results demonstrate that load demand response is closely linked to the islanding process, and an optimal fault recovery strategy can be achieved by utilizing the distributed energy storage system and the main network.

With the rise of digital technology, global cross-border information flows are driving significant growth in international digital commerce. This paper employs Meta-analysis to examine the impact of cross-border information flows on global trade competitiveness. It outlines the Meta-analysis paradigm, explores the relationship between data element valorization and trade competitiveness, and highlights the varying effects across different stages of the trade process. Using correlation coefficients as effect values, the study transforms and calculates data with the help of formulas and software to comprehensively analyze and test the relationship. The findings reveal rapid growth in China’s digital economy, expanding from 22.6 trillion yuan in 2016 to 51.9 trillion yuan in 2022, deeply influencing industrial structures. In global cross-border data flows, China and Russia exhibit tighter regulations, with China’s DSTRI value rising from 0.325 to 0.347 million USD, demonstrating that cross-border data flows significantly impact global trade competitiveness.

In the era of intelligent education, technology is reshaping traditional music education by enhancing teaching quality, optimizing curriculum design, and improving teacher resources. However, its redistributive effects remain underexplored. This study examines how intelligent education technology impacts resource distribution in music education, focusing on the context of music teacher certification. The research highlights the reform needs of music teacher education, including student-centered goals, improved teaching methods, and optimized curricula. It introduces a music intelligence system based on a radial basis function (RBF) neural network and evaluates its potential in promoting equitable resource distribution through interactive teaching. Findings reveal that intelligent education technology enhances student learning outcomes and music skills by enabling personalized learning paths and strengthening practical teaching. Experimental results confirm the system’s effectiveness in significantly improving students’ music grades, demonstrating its value in transforming music education.

In the modern era, the cultivation of foreign talents extends beyond the traditional enhancement of humanistic knowledge, with literature playing a pivotal role. Addressing the challenges posed by the “golden curriculum,” this study uses the “Selected British and American Stories” program as an example to explore a blended learning and sorting approach. Aligned with the Ministry of Education’s emphasis on “golden subjects,” the research formulates an implementation strategy for curriculum development. In the context of the Ministry’s promotion of the mixed funding program in 2019, the study highlights the necessity of guiding students to utilize the Internet for data-driven blended learning. By emphasizing active engagement, intrinsic motivation, and flexible learning approaches, the proposed strategy aims to enhance teaching quality and align with contemporary educational reform priorities. Furthermore, the paper underscores the significance of equitable teaching evaluation as a feedback mechanism, actively contributing to the overall improvement of teaching quality.

An injective coloring of a given graph \(G = (V, E)\) is a vertex coloring of \(G\) such that any two vertices with a common neighbor receive distinct colors. An \(e\)-injective coloring of a graph \(G\) is a vertex coloring of \(G\) in which any two vertices \(v, u\) with a common edge \(e\) (\(e \neq uv\)) receive distinct colors; in other words, any two end vertices of a path \(P_4\) in \(G\) achieve different colors. With this new definition, we want to take a review of injective coloring of a graph from the new point of view. For this purpose, we review the conjectures raised so far in the literature of injective coloring and \(2\)-distance coloring, from the new approach of \(e\)-injective coloring. Additionally, we prove that, for disjoint graphs \(G, H\), with \(E(G) \neq \emptyset\) and \(E(H) \neq \emptyset\), \(\chi_{ei}(G \cup H) = \max\{\chi_{ei}(G), \chi_{ei}(H)\}\) and \(\chi_{ei}(G \vee H) = |V(G)| + |V(H)|.\) The \(e\)-injective chromatic number of \(G\) versus the maximum degree and packing number of \(G\) is investigated, and we denote \(\max\{\chi_{ei}(G), \chi_{ei}(H)\} \leq \chi_{ei}(G \square H) \leq \chi_{2}(G)\chi_{2}(H).\) Finally, we prove that, for any tree \(T\) (\(T\) is not a star), \(\chi_{ei}(T) = \chi(T),\) and we obtain the exact value of the \(e\)-injective chromatic number for some specified graphs.

In the literature of algebraic graph theory, an algebraic intersection graph called the invariant intersection graph of a graph has been constructed from the automorphism group of a graph. A specific class of these invariant intersection graphs was identified as the \(n\)-inordinate invariant intersection graphs, and its structural properties has been studied. In this article, we study the different types of proper vertex coloring schemes of these \(n\)-inordinate invariant intersection graphs and their complements, by obtaining the coloring pattern and the chromatic number associated.

This paper examines how digital entertainment consumption drives China’s economic growth from multiple dimensions. Using panel data from 260 prefecture-level cities (2020–2022) and a multi-temporal double-difference method, the study finds that digital entertainment consumption significantly promotes economic growth, with a direct effect coefficient of 0.748. Robustness tests via the PSM-DID method confirm this effect, with a coefficient of 0.714, significant at the 5% level. In the low digital divide group, the regression coefficient is 6.325, while it is significantly lower in the high digital divide group, indicating that the digital divide weakens the effect. Heterogeneity analysis shows that enhancing consumer experience, generating new businesses, and boosting cultural influence positively impact growth. The findings provide insights for the sustainable development of the entertainment industry and the digital economy.

Financial frauds, often executed through asset transfers and profit inflation, aim to reduce taxes and secure credits. To enhance the accuracy and efficiency of accounting data auditing, this study proposes an anomaly detection scheme based on a deep autoencoder neural network. Financial statement entries are extracted from the accounting information system, and global and local anomaly features are defined based on the attribute values of normal and fraudulent accounts, corresponding to individual and combined anomaly attribute values. The AE network is trained to identify anomalies using account attribute scores. Results demonstrate classification accuracies of 91.7%, 90.3%, and 90.9% for sample ratios of 8:2, 7:3, and 6:4, respectively. The precision, recall, and F1 score reach 90.85%, 90.77%, and 90.81%, respectively. Training takes 95.81ms, with recognition classification requiring only 0.02ms. The proposed deep neural network achieves high recognition accuracy and speed, significantly improving the detection of financial statement anomalies and fraud.

The core of financial institutions’ big data lies in risk control, making network security threat identification essential for enhancing data processing and service levels. This study applies the principles of network information transmission security prevention, combining frequency domain analysis and distributed processing to extract threat characteristics. A financial network security threat identification model is developed using BiGRU and Transformer models, and a SQLIA defense system is constructed by integrating multi-variant execution and SQL injection attack prevention. Additionally, an intelligent network security defense strategy is formulated based on finite rationality theory. Simulation results show an F1 composite score of 90.78% for threat identification, and the STRIPS-BR defense strategy reduces relative risk by 74.81% during peak times compared to other strategies. Supported by big data, this system ensures secure data transmission and enhances the network service capabilities of financial institutions.

Fine chemical processes are integral to modern industries such as automotive, environmental protection, aviation, and new energy. However, these processes involve highly toxic substances and complex chemical interactions, making them vulnerable to uncontrollable circumstances and posing significant risks to human safety and the environment. This work proposes an enhanced GA-LVW algorithm for reliability assessment of fine chemical processes, focusing on essential operating units. The method utilizes global-local structure analysis to extract features from operating unit variables, reducing data noise, simplifying the construction of fuzzy rules, and improving model resilience. The extracted features are integrated into a fuzzy inference system. The proposed approach is validated using the Tennessee Eastman (TE) process model and the R-22 production process in a fluoride facility. Results demonstrate that the enhanced GA-LVW algorithm significantly improves the system’s efficiency and maintainability compared to conventional fuzzy inference systems.

Over the past two decades, with the support of the Party and the state, universities have established educational principles integrating curriculum reform, teaching beliefs, and political theories. Despite significant progress in ideological and political theory research, challenges remain that hinder sustainable development. This paper leverages a computerized algorithmic model of complex information networks to explore the intersection of scientific and humanistic approaches in education. By combining these methods, the study provides an optimized knowledge and political model for university education and analyzes its credibility. Empirical results indicate that the proposed model achieves a 91% accuracy rate. The improved model enhances the intellectual and political vitality of university theoretical courses, strengthens educational principles, and ensures the quality of university education.

A positive integer \(k\) is called a magic constant if there is a graph \(G\) along with a bijective function \(f\) from \(V(G)\) to the first \(|V(G)|\) natural numbers such that the weight of the vertex \(w(v) = \sum_{uv \in E} f(u) = k\) for all \(v \in V\). It is known that all odd positive integers greater than or equal to \(3\) and the integer powers of \(2\), \(2^{t}\), \(t \geq 6\), are magic constants. In this paper, we characterize all positive integers that are magic constants and generate all distance magic graphs, up to isomorphism, of order up to \(10\).

The Radenković and Gutman conjecture establishes a relationship between the Laplacian eigenvalues of any tree \(T_n\), the star graph \(S_n\), and the path graph \(P_n\), i.e., \({LE}(P_n) \leq {LE}(T_n) \leq {LE}(S_n).\) In this paper, we prove this conjecture for a class of trees with \(n\) vertices and having diameter \(16\) to \(30\).

To address large prediction errors in traditional risk assessment methods, the X-means clustering algorithm is utilized to segment financial product customers, combined with correlation strength analysis to understand customer behaviors and needs. Using the Hoteling model, a two-step pricing strategy is proposed, revealing that data product prices are inversely proportional to depreciation rate, timeliness, and customization degree, and deriving the platform’s optimal pricing strategy. A financial risk indicator system is developed using principal component analysis for systematic risk assessment. In call option pricing prediction, the model converges at Epoch=40, achieving a normalized predicted price of 0.154 (true value: 0.153). For put options, the model converges at Epoch=100, with a predicted normalized price of 0.146 (true value: 0.145). The results demonstrate the model’s accuracy in pricing prediction, providing effective support for real-time market risk monitoring and timely risk prevention.

This study develops a stereoscopic vision system using a two-camera calibration method and BP neural networks combined with genetic algorithms to measure precision component dimensions. Images are processed using edge detection and Hough transform algorithms, and a machine vision-based inspection model is constructed. Bearing components are used as the research object to detect dimensions, edges, geometric parameters, and loose components under six angles. Maximum measurement deviation is 0.04 mm, and edge detection results are clear and concise. Geometric parameter deviations remain within [-5%, 5%], achieving high recognition accuracy. The detection model’s classification accuracy is 97.49%, with verification accuracy at 98.01%. Comprehensive false detection and leakage rates are 1.03% and 0.46%, respectively. The model demonstrates superior detection performance across various angles for bearing components.

We study a discrete-time model for the spread of information in a graph, motivated by the idea that people believe a story when they learn of it from two different origins. Similar to the burning number, in this problem, information spreads in rounds and a new source can appear in each round. For a graph \(G\), we are interested in \(b_2(G)\), the minimum number of rounds until the information has spread to all vertices of graph \(G\). We are also interested in finding \(t_2(G)\), the minimum number of sources necessary so that the information spreads to all vertices of \(G\) in \(b_2(G)\) rounds. In addition to general results, we find \(b_2(G)\) and \(t_2(G)\) for the classes of spiders and wheels and show that their behavior differs with respect to these two parameters. We also provide examples and prove upper bounds for these parameters for Cartesian products of graphs.

This study explores how employee satisfaction moderates the relationship between corporate performance and innovative behavior using deep learning models: Autoencoder and restricted Boltzmann machines (RBM). The Autoencoder extracts key features for better analysis, while the RBM-based model analyzes the relationships among employee satisfaction, corporate performance, and innovative behavior. Results show a positive correlation between employee satisfaction and innovative behavior (0.460) and between innovative behavior and corporate performance (0.348). Regression analysis reveals that employee satisfaction indirectly impacts corporate performance through innovative behavior (impact: 0.10, t = 5.25). Differences in satisfaction, innovative behavior, and performance were observed across employee attributes. This study highlights the role of employee satisfaction in enhancing corporate performance and innovation, offering insights for human resource strategies.

An hourglass \(\Gamma_0\) is the graph with degree sequence \(\{4,2,2,2,2\}\). In this paper, for integers \(j\geq i\geq 1\), the bull \(B_{i,j}\) is the graph obtained by attaching endvertices of two disjoint paths of lengths \(i,j\) to two vertices of a triangle. We show that every 3-connected \(\{K_{1,3},\Gamma_0,X\}\)-free graph, where \(X\in \{ B_{2,12},\,B_{4,10},\,B_{6,8}\}\), is Hamilton-connected. Moreover, we give an example to show the sharpness of our result, and complete the characterization of forbidden induced bulls implying Hamilton-connectedness of a 3-connected {claw, hourglass, bull}-free graph.

Special attention has been given to China’s socio-economic development, the gradual improvement of living standards, and the increasing emphasis on preschool education by families and society. However, this process is influenced by various factors, such as school conditions, family dynamics, teacher performance, and social influences, which negatively affect the quality of kindergarten brand image and learning outcomes. These challenges hinder the effective empowerment of children across different fields. To achieve the goals of kindergarten education, teachers should leverage the comprehensive nurturing value of labor education to maximize and optimize its educational impact. Kindergarten brand image evaluation is a critical component of early childhood education, helping educators and researchers assess its effectiveness and identify areas for development. This paper addresses the issues in China’s current kindergarten brand image evaluation practices and proposes an evaluation method based on the support vector mechanism (SVM) and component analysis to enhance evaluation quality. The proposed approach aims to improve the accuracy and reliability of kindergarten brand image assessments, contributing to the advancement of early childhood education.

This research delves into the pathway energy framework for flower families, a class of simple connected graphs, whose path matrix \( P \) is constructed such that each entry \( P_{ij} \) quantifies the maximum number of vertex-disjoint paths. By analyzing the characteristic values of this matrix, we establish the pathway energy bounds specific to these flower graph families. Additionally, a comprehensive algorithm is developed to evaluate the time complexity across different flower family configurations, utilizing numerous trials to capture their average, maximum, and minimum computational behaviors. This analysis offers a comparative study of the structural intricacies that lead to increased computational complexity, highlighting which graph topologies tend to impose higher algorithmic challenges. The proposed method introduces a refined and adaptable approach, deepening the exploration of characteristic graph properties and their computational impact, thereby expanding the practical applications of these findings in graph theory.

Let \(G=(V,E)\) be a simple connected graph with vertex set \(V\) and edge set \(E\). The Randić index of graph \(G\) is the value \(R(G)=\sum_{uv\in E(G)} \frac{1}{\sqrt{d(u)d(v)}}\), where \(d(u)\) and \(d(v)\) refer to the degree of the vertices \(u\) and \(v\). We obtain a lower bound for the Randić index of trees in terms of the order and the Roman domination number, and we characterize the extremal trees for this bound.

This study investigates the impact of gamification teaching on students’ motivation in physical education using questionnaires, teaching experiments, and mathematical statistics. A gamified sports teaching model, grounded in the self-determination motivation theory and analyzed through a multiple regression model, was designed to assess motivational stimulation. Results showed that gamified physical education significantly improved motivation in the experimental class compared to the control class (P < 0.05). The average physical education score in the experimental class was 77.67, 5.08 points higher than the control class. Internal motivation, identity regulation, intake regulation, and external regulation ratings were 4.132, 3.992, 4.172, and 4.156, respectively. Regression analysis confirmed that gamified teaching positively influenced motivation, with self-determination theory effectively mediating students’ physical education learning motivation.

In this paper, it is pointed out that the definition of `Fibonacci \((p,r)\)-cube’ in many papers (denoted by \(I\Gamma_{n}^{(p,r)}\)) is incorrect. The graph \(I\Gamma_{n}^{(p,r)}\) is not the same as the original one (denoted by \(O\Gamma_{n}^{(p,r)}\)) introduced by Egiazarian and Astola. First, it is shown that \(I\Gamma_{n}^{(p,r)}\) and \(O\Gamma_{n}^{(p,r)}\) have different recursive structure. Then, it is proven that all the graphs \(O\Gamma_{n}^{(p,r)}\) are partial cubes. However, only a small part of graphs \(I\Gamma_{n}^{(p,r)}\) are partial cubes. It is also shown that \(I\Gamma_{n}^{(p,r)}\) and \(O\Gamma_{n}^{(p,r)}\) have different medianicity. Finally, several questions are listed for further investigation.

A \(q\)-total coloring of \(G\) is an assignment of \(q\) colors to the vertices and edges of \(G\), so that adjacent or incident elements have different colors. The Total Coloring Conjecture (TCC) asserts that a total coloring of a graph \(G\) has at least \(\Delta+1\) and at most \(\Delta+2\) colors. In this paper, we determine that all members of new infinite families of snarks obtained by the Kochol superposition of Goldberg and Loupekine with Blowup and Semiblowup snarks are Type~1. These results contribute to a question posed by Brinkmann, Preissmann and D. Sasaki (2015) by presenting negative evidence about the existence of Type~2 cubic graphs with girth at least 5.

Generative adversarial network (GAN) technology has enabled the automatic synthesis of realistic face images from text. This paper proposes a model for generating face images from Chinese text by integrating a text mapping module with the StyleGAN generator. The text mapping module utilizes the CLIP model for pre-training Chinese text, employs a convolutional-inverse convolutional structure to enhance feature extraction, and incorporates a BiLSTM model to construct complete sentences as inputs for the StyleGAN generator. The generator interprets semantic features to generate face images. Validation on Face2Text and COCO datasets yields F1 values of 83.43% and 84.97%, respectively, while achieving the lowest FID and FSD scores of 103.25 and 1.26. The combination of CLIP pre-training and word-level semantic embedding improves image quality, offering a novel approach for face recognition applications in public safety.

In this note, we establish six Gallai theorems involving twelve minority and majority parameters. Accordingly, the complexity problems corresponding to some of these parameters are obtained.

The promotion of industrial digital transformation is a crucial breakthrough in the evolution of economic structures and the physical layout of spaces. It has the potential to elevate the entire industrial chain to a high-end value chain, creating more profit opportunities and enhancing the influence of domestic industries in the international cycle. This study uses the cities in the Yangtze River Delta Economic Belt as a case study to explore the spatial effects of digital transformation on the healthy transformation of traditional industrial structures. It constructs relevant spatial coupling models and empirically verifies them by testing specific assumptions. The experimental results indicate that the model is significant at a level greater than 5%, making it suitable for selecting spatial measurement models. The mean square error of its network simulation output is 0.1333, confirming the expected hypothesis and demonstrating that digital transformation has a significant spatial driving effect on industrial upgrading.