Algebraic Integers as Chromatic and Domination Roots. (14th May 2012)

• Record Type: Journal Article
• Authors: Alikhani, Saeid; Hasni, Roslan
• Other Names: Li Xueliang Academic Editor.
• Is Part Of: International journal of combinatorics. Volume 2012(2012)
• Publication Date: 2012-05-14
• Subjects: Combinatorial analysis -- Periodicals; Combinatorial analysis; Periodicals; Electronic journals; 511.605
• DOI: 10.1155/2012/780765
• Languages: English
• ISSNs: 1687-9163

Abstract

Abstract : Let G be a simple graph of order n and λ ∈ ℕ . A mapping f : V ( G ) → { 1, 2, …, λ } is called a λ -colouring of G if f ( u ) ≠ f ( v ) whenever the vertices u and v are adjacent in G . The number of distinct λ -colourings of G, denoted by P ( G, λ ), is called the chromatic polynomial of G . The domination polynomial of G is the polynomial D ( G, λ ) = ∑ i = 1 n d ( G, i ) λ i, where d ( G, i ) is the number of dominating sets of G of size i . Every root of P ( G, λ ) and D ( G, λ ) is called the chromatic root and the domination root of G, 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.