BIPARTITE EDGE FRUSTRATION OF SOME NANOTUBES

Let G = (V,E) be a simple graph, a graph without multiple edges and loops. A subgraph S of G is a graph whose set of vertices and set of edges are all subsets of G. A spanning subgraph is a subgraph that contains all the vertices of the original graph. The graph G is called bipartite if the vertex set V can be partitioned into two disjoint subsets V1 and V2 such that all edges of G have one endpoint in V1 and the other in V2. Bipartite edge frustration of a graph G, denoted by φ(G), is the minimum number of edges that need to be deleted to obtain a bipartite spanning subgraph. It is easy to see that φ(G) is a topological index and G is bipartite if and only if φ(G) = 0. Thus φ(G) is a measure of bipartivity. It is a well-known fact that a graph G is bipartite if and only if G does not have odd cycles. Holme, Liljeros and Edling introduced the edge frustration as a measure in the context of complex network, [8]. In [5,6] Fajtlowicz claimed that the chemical stability of fullerenes is related to the minimum number of vertices/edges that need to be deleted to make a fullerene graph bipartite. We mention here that before publishing the mentioned papers of Fajtlowicz, Schmalz et al. [10] observed that the isolated pentagon fullerenes (IPR fullerenes) have the best stability. Doslic [1], presented some computational results to confirm this relationship. So it is natural to ask about relationship between the degree of non-bipartivity of nanotubes and their stability. Throughout this paper all graphs considered are finite and simple. Our notation is standard and taken mainly from [7,9]. We encourage the reader to consult papers by Doslic [1-4] for background material and more information on the problem. Our main results are the following two theorems: Theorem 1. Suppose E = TUC4C8(R)[p,q], Figure 1, and F = TUC4C8(S)[p,q], Figure 2, are C4C8 nanotubes in which p and q are the number of rhombs and squares in each row and column, respectively. Then φ(F) =0 and