Computing the anti-Kekulé number of certain nanotubes and nanocones

Let G(V,E) be a connected graph. A set M subset of E is called a matching if no two edges in M have a common end-vertex. A matching M in G is perfect if every vertex of G is incident with an edge in M. The perfect matchings correspond to Kekulé structures which play an important role in the analysis of resonance energy and stability of hydrocarbons. The anti-Kekulé number of a graph G, denoted as ak(G), is the smallest number of edges which must be removed from a connected graph G with a perfect matching, such that the remaining graph stay connected and contains no perfect matching. In this paper, we calculate the anti-Kekulé number of TUC₄C₈(S)[p,q] nanotube, TUC₄C₈(S)[p,q] nanotori for all positive integers p, q and CNC_2k-1[n] nanocones for all positive integers k and n.