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.