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Graph theory as a branch of discrete mathematics has experienced significant development in its application to modern complex network systems, particularly in digital social networks and transportation systems. This research aims to analyze fundamental concepts of graph theory, examine characteristics of cycle detection algorithms along with their computational complexity, investigate their application in digital social network analysis, and explore their implementation in digital transportation system optimization. The research method employs a qualitative approach with library research focusing on scientific literature from 2020-2025 period from accredited academic databases such as Scopus, Web of Science, and IEEE Xplore, utilizing thematic analysis techniques to identify meaningful patterns from the examined literature. Research findings indicate that fundamental graph theory concepts including vertices, edges, and graph classifications form the foundation for relational structure modeling. Cycle detection algorithms such as Depth-First Search, Union-Find, and Tarjan demonstrate effectiveness with O(V+E) complexity for large-scale graphs. Applications in digital social networks facilitate community identification through Multi-View Clustering, centrality analysis for influencer detection, and understanding viral information dissemination patterns. Implementation in digital transportation systems demonstrates route planning optimization using Dijkstra and Bellman-Ford algorithms, vulnerability analysis through articulation point and bridge identification, and bottleneck detection with betweenness centrality. The research concludes that integration of graph theory in discrete mathematics education enhances critical thinking skills and real-world application understanding, with recommendations for algorithm development for massive dynamic graphs and machine learning integration in graph algorithm optimization.

Learning is the process of children getting to know knowledge that aims to increase knowledge as a basis for forming behavior. The learning process is carried out so that children experience behavioral changes and gain an overall improvement in knowledge, skills, and attitudes. Learning is very closely related to daily life, therefore it is important for children to learn continuously, one of which is through learning mathematics. Mathematics learning has a very important role in the sustainability of children's education because mathematics is one of the fundamental non-physical tools in the world of education and daily life. Through mathematics learning, children are trained to think logically, systematically, critically, and creatively in solving various problems. In addition, mathematics helps children solve calculation problems that are often encountered in daily activities, such as counting money, measuring length, time, and distance, and interpreting simple data. Mathematics learning also plays a role in developing the ability to reason and make appropriate decisions. Therefore, mathematics learning is a subject that must be followed because it has a great influence on thinking skills, learning independence, and children's readiness to face life challenges and scientific development in the future.

Dyscalculia is a difficulty in learning mathematics that is commonly experienced by elementary school students, especially in multiplication materials. Students with dyscalculia experience obstacles in understanding the concept of numbers and calculation operations, which has an impact on low learning outcomes and students' confidence in mathematics learning. This study aims to describe the difficulties of learning multiplication in dyscalculia students in grade III of MI Nurul Falah 3, as well as identify learning efforts that can help overcome these difficulties. This study used a qualitative approach with a case study design on two students who were identified as having dyscalculia. Data collection techniques include observation, diagnostic assessment, interviews with students and teachers, and analysis of learning outcomes. The results showed that students had difficulty recognizing number symbols, often flipped numbers, and were not able to complete simple multiplication independently. Concrete learning using the "Multiplication Smart Wallet" media has been proven to help students understand the concept of multiplication, increase learning motivation, and reduce anxiety about mathematics. Thus, the use of concrete media is considered effective in supporting more inclusive mathematics learning for dyscalculia students.

This article discusses community service aimed at improving students' mathematical understanding at MI Nurul Mun'im through learning assistance assisted by the Astromath game. The main problem faced is low student motivation and understanding in mathematics, caused by less engaging learning methods. The purpose of this activity is to provide a solution by using innovative educational technology, specifically the Astromath game, which is designed to develop students' mathematical abilities in an interactive and fun way. The method used in this community service is a direct mentoring approach that involves students in game-assisted learning sessions, with mentoring from lecturers and students. This activity was carried out over several meetings involving the direct use of the Astromath game in class. The results obtained showed an increase in student understanding of mathematics material, especially in terms of basic arithmetic operations, as well as a significant increase in learning motivation. This is indicated by an increase in the average pre-test score from 58.8 to 88.9 average post-test score. In addition, students were also more active and interested in participating in mathematics lessons after being involved in learning sessions using this game. The use of the Astromath game has proven effective in making mathematics learning more interesting and improving student learning outcomes at MI Nurul Mun'im.

This Community Partnership Program (PKM) aims to improve the competence of mathematics teachers at Madrasah Tsanawiyah (MTs) in Lumajang Regency in developing ethnomathematics-based teaching materials integrated with Augmented Reality (AR) technology. This program was initiated due to the minimal use of local culture in mathematics learning and the low ability of teachers to develop innovative teaching materials. The activity was carried out in four main stages: socialization, training, technology application, and mentoring and evaluation. The training included an introduction to the concept of ethnomathematics, the preparation of differentiated teaching materials, and the creation of AR-based learning media using Unity 3D, Blender, Geogebra, and Vuforia. The results of the activity showed an increase in teachers' understanding of the concept of ethnomathematics and their ability to integrate the local cultural values of Lumajang into mathematics learning. In addition, teachers successfully produced contextual AR-based interactive teaching materials in line with the spirit of the Merdeka Curriculum. This activity also strengthened collaboration between lecturers, students, and teachers of the MTs Wilker III Mathematics MGMP and became a model for continuous training in the development of innovative learning media based on local wisdom and digital technology.

The purpose of this study was to examine how problem-based mathematics learning affected the learning interest of 5- to 6-year-old students at the Ki Hajar Dewantara State Kindergarten in South Jakarta. To compare problem-based mathematics learning and conventional learning, this study used a quantitative method and a quasi-experimental design. Problem-based learning is intended to use mathematical problems in children's daily lives. Children in group B were divided into experimental and control groups. To collect data, structured observations were conducted on elements of children's interest in learning. These included enjoyment, focus, activity, and willingness to participate in learning activities. To determine the difference in the level of interest in learning between the two groups, the data obtained was analyzed statistically. The results showed that, compared to conventional learning, problem-based mathematics learning had a significant effect on increasing children's interest in learning. Children showed more enthusiasm, better concentration, active involvement in activities, and critical thinking skills in solving mathematical problems. The results showed that effective problem-based mathematics learning made learning meaningful, enjoyable, and appropriate for the characteristics of early childhood development.

The application of Mastery Learning in Olympiad mentoring at Darussalam Islamic Junior High School was carried out using a classroom action research model involving 20 students from grades 8 and 9. The mentoring process followed a cyclical structure consisting of planning, action, observation, and reflection, supported by observation techniques, tests, and documentation. The implementation of the mastery learning model effectively improved students’ mathematics learning outcomes, as shown by a significant increase in mastery levels: 65% before the intervention, 75% after the first cycle, and 90% after the second cycle. In addition, the average learning score rose from 71 in the first cycle to 87 in the second cycle, reflecting substantial progress in understanding and problem-solving abilities. These results indicate that Mastery Learning not only enhances students’ mathematics performance but also strengthens their readiness and confidence to compete in the National OSN competition, demonstrating the model’s positive contribution to academic achievement and Olympiad preparation.

This study investigates the impact of GeoGebra-assisted collaborative learning on students' understanding of function graphs. Function graphs are fundamental in mathematics education, yet many students struggle to grasp the relationships between variables, primarily due to traditional teaching methods that focus on procedural skills rather than conceptual understanding. To address this challenge, the study incorporates GeoGebra, a dynamic mathematics software, alongside collaborative learning strategies. The research utilizes a quasi-experimental design involving high school students who had previously struggled with function graphs. The results demonstrate that the experimental group, which engaged in GeoGebra-assisted collaborative learning, showed a significant improvement of 27% in their post-test scores, compared to just a 6% improvement in the control group using traditional methods. The study highlights the effectiveness of GeoGebra in fostering a deeper conceptual understanding of mathematical functions by enabling students to visualize and manipulate graphs interactively. Additionally, collaborative learning encouraged peer interaction, reinforcing the learning process and promoting better problem-solving skills. The findings suggest that combining interactive tools like GeoGebra with collaborative learning techniques can enhance students’ mathematical comprehension, leading to improved engagement and performance in mathematics education.