On the degeneracy of the orbit polynomial and related graph polynomials

Abstract

The orbit polynomial is a new graph counting polynomial which is defined as (formula presented) where O1, …, Or are all vertex orbits of the graph G. In this article, we investigate the structural properties of the automorphism group of a graph by using several novel counting polynomials. Besides, we explore the orbit polynomial of a graph operation. Indeed, we compare the degeneracy of the orbit polynomial with a new graph polynomial based on both eigenvalues of a graph and the size of orbits.