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.