Abstract
In this article, we survey the results on examining orbit structures combined with polynomials, automorphism groups, roots of polynomials, and the construction of graphs with prescribed orbit structures. The orbit polynomial has been defined as the \(
\sum_{n} cx^n \), where \( c \) represents the number of orbits of graph \( G \) with size \( n \). By subtracting this polynomial from 1, the modified orbit polynomial \( O_G^{\star}(x) = 1 - O_G(x) \) is obtained which possesses a unique
positive root denoted by \( \delta \). This root can be seen as a relative measure of symmetry. The study of orbit structures in graphs and their associated automorphism groups is a fundamental topic in graph theory. The understanding
and analysis of these structures provide valuable insights into the symmetries in graphs, enabling the exploration of various graph properties and their applications in diverse fields such as network analysis, computer science, and chemistry.
