Using Tree Graphs to Solve Some Problems in Combinatorial Analysis

Abstract

This paper aims to explore computational methods for solving non-repetitive and repetitive permutations using tree graphs as tools for representation and problem-solving in combinatorial analysis. The study aims to assess the effectiveness of tree graphs in addressing these problems and to identify their limitations in cases involving repetitive elements. The study employs a systematic review, sourcing literature from reputable databases like Web of Science, Scopus, and IEEE and search engines such as Google Scholar. Specific keywords related to permutations, graphs, and generating functions were utilized to manage and categorize the retrieved articles. The results demonstrate the effectiveness of tree graphs in modeling non-repetitive permutations by representing the stages of the permutation process step-by-step. However, repetitive permutations, characterized by identical element groups, cannot be effectively represented using traditional tree graphs. The findings show that homogeneous complete graphs with specific stages are required for repetitive permutations. Additionally, the study reveals that tree graphs offer significant computational advantages in solving non-repetitive permutation problems, especially in cryptography, optimization, and resource allocation. While effective for non-repetitive cases, alternative methods, such as generating functions, are recommended for handling repetitive permutations. The paper provides valuable insights into applying tree graphs for combinatorial analysis, highlighting their strengths and limitations in various computational contexts.