Digest Journal of Nanomaterials and Biostructures Vol. 6, No 4, October-December 2011, p. 1551-1556 CHROMATIC POLYNOMIALS OF CERTAIN FAMILIES OF DENDRIMER NANOSTARS NABEEL E. ARIF, ROSLAN HASNI * N ∈ λ , SAEID ALIKHANI a School of Mathematical Sciences, Universiti Sains Malaysia 11800 USM, Penang, Malaysia a Department of Mathematics, Yazd University, 89175-741, Yazd, Iran Let G be a simple graph and . A mapping } , {1,2, ) ( : λ  → G V f is called a λ -colouring of G if ) ( ) ( v f u f ≠ whenever the vertices u and v are adjacent in G. The number of distinct λ -colourings of G , denoted by ) , ( λ G P is called the chromatic polynomial of G. A dendrimer is an artificially manufactured or synthesized molecule built up from branched units called monomers. In this paper, using the chromatic polynomial of some specific graphs, we compute the chromatic polynomials for certain families of dendrimer nanostars. (Received June 6, 2011; accepted September 20, 2011) Keywords: Chromatic polynomial, Dendrimer, Graph 1. Introduction A simple graph ) , ( = E V G is a finite nonempty set ) (G V of objects called vertices together with a (possibly empty) set ) (G E of unordered pairs of distinct vertices of G called edges. In chemical graphs, the vertices of the graph correspond to the atoms of the molecule, and the edges represent the chemical bonds. Let ) , ( λ χ G denotes the number of proper vertex colourings of G with at most λ colours. G. Birkhoff 4 , observed in 1912 that ) , ( λ χ G is, for a fixed graph , G a polynomial in λ , which is now called the chromatic polynomial of G . More precisely, let G be a simple graph and N ∈ λ . A mapping } , {1,2, ) ( : λ  → G V f is called a λ -colouring of G if ) ( ) ( v f u f ≠ whenever the vertices u and v are adjacent in . G The number of distinct λ -colourings of , G denoted by ) , ( λ G P is called the chromatic polynomial of . G The book by F.M. Dong, K.M. Koh and K.L. Teo 5 gives an excellent and extensive survey of this polynomial. A topological index is a real number related to a graph. It must be a structural invariant, i.e., it is fixed by any automorphism of the graph. There are several topological indices have been defined and many of them have found applications as means to model chemical, pharmaceutical and other properties of molecules. The Wiener index W and diameter are two examples of topological indices of graphs (or chemical model). For a detailed treatment of these indices, the reader is referred to 9,10. Dendrimers are hyper-branched macromolecules, with a rigorously tailored architecture. They can be synthesized, in a controlled manner, either by a divergent or a convergent procedure. Dendrimers have gained a wide range of applications in supra-molecular chemistry, particularly in host guest reactions and self-assembly processes. Their applications in chemistry, biology and nano-science are unlimited. Recently, some people investigated the mathematical properties of * Correspondence author: hroslan@cs.usm.my