The Anti-Kekulé Number And Anti-forcing Number Of Fullerenes

The anti-Kekule number of a connected graph G is the smallest number of edges that are deleted from G such that the remaining graph is still connected but it has no Kekule structures. In this paper, we show that the anti-Kekule number of any leapfrog fullerene is 4 in the base of results shown in [12]. Furthermore, by the cyclically edge-connectivity of fullerene graphs and Tutte's Theorem, we prove that the anti-Kekule number of any fullerene is 4, too.The anti-forcing number of a graph G with at least one perfect matching is the smallest number of edges that are deleted from G such that the remaining graph has only one perfect matching. In this paper, we obtain the lower bound of the anti-forcing number of fullerenes. Then we show this lower bound is sharp. Furthermore, we prove that there exists at least one fullerene which achieve the lower bound for any n≥20 (n≠22,26).