It is well known that determining the exact values of crossing number for circulant graphs is very difficult. Even so, some important results in this field are still proved. D.J. Ma was proved that the crossing number of C(2m + 2, m) is m + 1[8]. Then such problem for C(n, 3) was further solved [7]. Pak Tung Ho and X. Lin obtained accurate values for the crossover numbers of C (3m, m) and C (3m + 1, m)[4][5]. In this paper, as a complement, we show that the edges from the principal cycle of C(9, 3) do not cross each other in an optimal drawing.

The continuous development of emerging economies represented by China and Russia has exacerbated the changes in the world political and economic landscape, and international organizations represented by the United Nations have led to inefficient dispute resolution mechanisms in international affairs due to their formalism and pluralism. On the contrary, G-groups has shown its flexibility and efficiency in global governance. However, the international community has been questioned G-group’s legitimacy for many years. This paper will take the G7 and G20 as examples, analyze the legitimacy problems in G-groups, explain their reform measures, and propose future reform directions to promote the development of G-groups, so as to help the international community to conduct global governance more effectively.

Using the Resource Advantage Theory approach, this research aims to examine the gap between entrepreneurial opportunities and marketing performance, with market-based innovation capability acting as a mediating variable. The data collection method used non-probability sampling with a purposive sampling technique. The data that was eligible to be processed were 250 respondents. Hypothesis testing was used using the AMOS application. The research results show that market-based innovation capability can improve marketing performance as a mediating variable. In addition, market penetration strength can also improve marketing performance. As a strategic variable, market-based innovation capability (MBIC) converts entrepreneurial opportunities into competitive advantages relevant to market needs. In addition, business actors become more adaptive and responsive to market dynamics, increasing competitiveness sustainably. MBIC, rooted in the Resource Advantage Theory of competition, contributes to developing market-based innovation strategies in the UMKM sector.

Objective: To explore the influencing factors of mental health and the mediating role of self-compassion between family cohesion and mental health. Method: Family Cohesion Scale, Symptom Checklist, and Self-compassion Scale were used to investigate 593 college students in Zhejiang Province. Result: Family Cohesion was negatively correlated with mental health and positively correlated with Self-compass- ion among college students; Self-compassion was negatively correlated with mental health. Self-compassion fully mediates the relationship between the two. Conclusion: The path of family cohesion is indirect, and strengthening Self-compassion education can improve the mental health level of college students.

This study analyzes the perception of university students regarding the use of virtual reality (VR) in higher education, focusing on their level of knowledge, usage, perceived advantages and disadvantages, as well as their willingness to use this technology in the future. Using a mixed-methods approach that combines questionnaires and semi-structured interviews, both quantitative and qualitative data were collected to provide a comprehensive view of the subject. The results indicate that while students have a basic understanding of VR, its use in the educational context is limited. A considerable number of students recognize VR’s potential to enhance the learning experience, particularly in terms of immersion and engagement. However, significant barriers to adoption were identified, such as technical issues, the high cost of equipment, and inadequate access to technological infrastructure. Additionally, there is a need for broader training for both students and faculty to ensure the effective use of this technology in academic environments. The semi-structured interviews confirmed that perceptions of VR vary depending on prior exposure to the technology and access to resources. Despite the challenges, most students appreciate VR’s potential to enrich learning, although its effective adoption will depend on overcoming the identified barriers. The study concludes that strategies must be implemented to facilitate the integration of VR into higher education, thus optimizing its impact on the teaching-learning process.

Employment guidance is an important way for employment education in colleges and universities, and it is also an important educational process. Strengthening the employment guidance in colleges and universities is necessary to alleviate the structural contradiction of employment and serve the talent cultivation of schools, as well as to enhance the employability of college students and serve the growth and success of young students. This paper focuses on the problems of employment guidance in colleges and universities in the new era, and proposes countermeasures to solve them in four dimensions: system construction, precise service, regional coordination and employment capacity enhancement.