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.

Through the research on the communication mode of rural youth e-commerce, the paper puts forward that there are some advantages and many problems in the e-commerce engaged in.

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.

As a group of college students who are about to enter social life and have a certain impact on socio-economic and technological development, their concepts and qualities will have a significant impact on social development. Therefore, in a complex and ever-changing social environment, schools should pay attention to educating college students in aspects such as their values, high moral literacy, and political awareness. Schools can take ideological and political education as the foundation, comprehensively educate students through the construction of a good academic atmosphere, and cultivate them into application-oriented high-quality talents. Based on this, this article mainly studies the construction methods of college students' academic style from the perspective of ideological and political education in the new era.

The purpose of the current study is to examine the mediating role of intercultural communicative competence on the relationship between teaching of English language and learning at Chinese higher vocational colleges. The convenience sampling technique was used to collect data from 668 teachers, teaching English language subjects in different public and private Chinese higher vocational colleges. Smart partial least squares-structural equation modeling on SmartPLS software version 4 was used to test the hypotheses. The result revealed the direct effect of English language teaching (ELT) is not significant on English language learning (ELL). However, the intercultural communicative competences (ICC) have been tested and proved to be a potential mediator between English language teaching and learning. Because the indirect effect of ELT on ELL is positive and significant through mediator ICC. Therefore, based on the findings of this study, it can be concluded that the inclusion of intercultural communication ability is a crucial component in the vocational education of college students. Policymakers should be cautious about promoting and expanding the availability of cultural teaching and learning across demographic conditions (e.g., linguistic and ethnic diversity, age, and gender) and various levels of language proficiency. In accordance with the effects of teacher education and professional development programs, the implementation of ICC content necessitates a harmonization of pedagogical approaches and assessment practices across designated levels in order to effectively achieve educational objectives. To promote ICC in English language education, there must be clear guidelines and communication to school leaders, educators, and administrators regarding the necessity and goals of cultural integration.

Lei's work explores the theme of existentialism, reflecting the anxiety and uncertainty that modern people experience in the face of a rapidly changing world. The stories explore the dislocation and suffering experienced by modern people in a heterogeneous cultural space, the anxiety of existential existence, and the alienation and dislocation of interpersonal relationships.