Network Analyzing by the Aid of Orbit Polynomial

Abstract

This article aims to be a further contribution to the research on structural complexity networks. Here, we emphasize measures to determine symmetry. The so-called “orbit polynomial” is defined by OG(x)=∑iaixi, where ai is the number of orbits of size i. Furthermore, the graph polynomial 1−OG(x) has a unique positive root in the interval (0,1), which can be considered as a relevant measure of the symmetry of a graph. In the present paper, we studied some properties of the orbit polynomial with respect to the stabilizer elements of each vertex. Furthermore, we constructed graphs with a small number of orbits and characterized some classes of graphs in terms of calculating their orbit polynomials. We studied the symmetry structure of well-known real-world networks in terms of the orbit polynomial.