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The orbit polynomial is a new graph counting polynomial which is defined as OG(x)=∑i=1rx|Oi|, 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.

On the Degeneracy of the Orbit Polynomial and Related Graph Polynomials