International Journal of Combinatorics

Volume 2012 (2012), Article ID 780765, 8 pages

http://dx.doi.org/10.1155/2012/780765

## Algebraic Integers as Chromatic and Domination Roots

^{1}Department of Mathematics, Yazd University, Yazd 89195-741, Iran^{2}Department of Mathematics, Faculty of Science and Technology, University Malaysia Terengganu, 21030 Kuala Terengganu, Malaysia

Received 19 January 2012; Accepted 7 March 2012

Academic Editor: Xueliang Li

Copyright © 2012 Saeid Alikhani and Roslan Hasni. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

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.