On The Anti-Kekulé Number And Anti-forcing Number Of Some Hexagonal Systems

The anti-forcing number and anti-Kekule number are introduced for studying Kekule structures in a benzenoid. Let G= (V, E) be a graph G with a perfect matching. An anti-forcing set of G is a subset A of E such that G-A has a unique Kekule structure. An anti-forcing set of the smallest cardinality is called a minimal anti-forcing set, and its cardinality is the anti-forcing number of G and it is denoted by af(G). An anti-Kekule set of G is the set S of E such that G-S is a connected graph and it has no Kekule structures. An anti-Kekule set of the smallest cardinality is called a minimal anti-Kekule set, and its cardinality is the anti-Kekule number of G and it is denoted by ak(G).In this paper, we prove the anti-Kekule numbers of hexagonal spiders is 2 or 3, and give an upper bound of anti-forcing number of hexagonal spiders and a algorithm for computing the anti-forcing number of hexagonal spiders S(n1,n2,n3) in which the length of each segment is more than 2. The calculation of these in-variants is demonstrated on rectangular models and skew strip models in this paper by analyzing the structures of their graphs, and it is shown that the anti-Kekule numbers of rectangular model R[k,l] and skew strip model Z[k, l] with l rows and k columns are 2, the anti-forcing number of R[k,l] is k and the anti-forcing number of Z[k,l] is not more than [(l+1/2)], where [x] is the greatest integer no more than x.