Chromatic Polynomials of Some Nanotubes

Let ) , ( λ χ G denotes the number of proper vertex colourings of with at most G λ colours. G. Birkhoff , observed in 1912 that ) , ( λ χ G is, for a fixed graph , a polynomial in G λ , which is now called the chromatic polynomial of . More precisely, let G be a simple graph and G N ∈ λ . A mapping } , {1,2, ) ( : λ K → G V f is called a λ -colouring of if whenever the vertices and are adjacent in G . The number of distinct G ) (v ) ( f u f ≠ u v λ -colourings of G , denoted by ) , ( λ G P is called the chromatic polynomial of . A zero of G ) , ( λ G P is called a chromatic zero of G . The book by F.M. Dong, K.M. Koh and K.L. Teo gives an excellent and extensive survey of this polynomial and its root. 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 .