Abstract
Suppose ai indicates the number of orbits of size i in graph G. A new counting polynomial, namely an orbit polynomial, is defined as OG(x) = ∑i aixi. Its modified version is obtained by subtracting the orbit polynomial from 1. In the present paper, we studied the conditions under which an integer polynomial can arise as an orbit polynomial of a graph. Additionally, we surveyed graphs with a small number of orbits and characterized several classes of graphs with respect to their orbit polynomials.
Highlights
By having the orbits and their structures in a graph, we can infer many algebraic properties about the automorphism group and about the similar vertices
If there exists a property that does not hold for two vertices, these vertices are not in the same orbit
Following the methods of [22], we introduce several classes of graphs that can be characterized by their orbit polynomials
Summary
By having the orbits and their structures in a graph, we can infer many algebraic properties about the automorphism group and about the similar vertices. The authors indicated that the unique positive root of this new polynomial can be served as a relative measure of a graph’s symmetry The magnitude of this root measures symmetry and can be used to compare graphs with respect to this property. In [22], several properties of orbit polynomial with respect to the stabilizer elements of each vertex and many classes of graphs with a small number of orbits were studied. This method was applied to investigate the symmetry structure of some real-world networks.
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