%0 Journal Article
%T Algebraic Integers as Chromatic and Domination Roots
%A Saeid Alikhani
%A Roslan Hasni
%J International Journal of Combinatorics
%D 2012
%I Hindawi Publishing Corporation
%R 10.1155/2012/780765
%X Let be a simple graph of order and . A mapping 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 , 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. 1. Introduction Let be a simple graph and . A mapping 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 . A zero of is called a chromatic zero of . For a complete survey on chromatic polynomial and chromatic root, see [1]. For any vertex , the open neighborhood of is the set , and the closed neighborhood is the set . For a set , the open neighborhood of is , and the closed neighborhood of is . A set is a dominating set if , or, equivalently, every vertex in is adjacent to at least one vertex in . An -subset of is a subset of of cardinality . Let be the family of dominating sets of which are -subsets and let . The polynomial is defined as domination polynomial of [2, 3]. A root of is called a domination root of . We denote the set of all roots of by . For more information and motivation of domination polynomial and domination roots, refer to [2¨C6]. We recall that a complex number is called an algebraic number (respectively, algebraic integer) if it is a zero of some monic polynomial with rational (resp., integer) coefficients (see [7]). Corresponding to any algebraic number , there is a unique monic polynomial with rational coefficients, called the minimal polynomial of (over the rationals), with the property that divides every polynomial with rational coefficients having as a zero. (The minimal polynomial of has integer coefficients if and only if is an algebraic integer.) Since the 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 Sections 2 and 3, we study algebraic integers as chromatic roots and domination roots,
%U http://www.hindawi.com/journals/ijcom/2012/780765/