Algebraic Integers as Chromatic and Domination Roots

Let be a simple graph of order and ∈？. A mapping ∶()→{1,2,…,} is called a -colouring of if ()≠() whenever the vertices and are adjacent in . The number of distinct -colourings of , denoted by (,), is called the chromatic polynomial of . The domination polynomial of is the polynomial ∑(,)==1(,), where (,) is the number of dominating sets of of size . Every root of (,) and (,) is called the chromatic root and the domination root of , respectively. Since chromatic polynomial and domination polynomial are monic polynomial with integer coefficients, its zeros are algebraic integers. This naturally raises the question: which algebraic integers can occur as zeros of chromatic and domination polynomials? In this paper, we state some properties of this kind of algebraic integers.