A Study of Roots of a Certain Class of Counting Polynomials

Abstract

Suppose G is a graph with vertex-orbits O1,1 O2, . . . ,Ot, and jOi j denotes the cardinallity
 of Oi. Then OG (x) = åt
 2 i=1 xjOi j is called as orbit polynomial. It is well-known that this polynomial
 3 has a unique positive zero d in the interval [0, 1]. The aim of this paper is to study the specific
 4 properties of this polynomial and then we determine the location of this root for several classes of
 5 complex networks to compare with other graphical measures. Besides, we compare the unique
 6 positive zero measure with several well-known centrality graph measures.