On the anti-kekulé number of nanotubes and nanocones

Let G(V,E)be a connected graph. A set M ⊆ 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 akG, 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. The anti-Kekulé numbers of silicate, oxide and honeycomb networks were calculated in [Xavier, Shanthi, and Raja, International Journal of Pure and Applied Mathematics 6, 1019 (2013)]. In this paper, we calculate the anti-Kekulé number of HAC5C7/2p(q), TUC4(C8)R,2p(q), ∀ pq nanotubes and CNC2kn, ∀ k(n) nanocones. We set the infinite cases of all nanotubes as conjecture.