Zhang-Zhang Polynomials And Anti-forcing Number Of Cyclo-polyphenacenes

Let G=(V,E) be a simple connected graph.A perfect matching(or Kekul(?) structure) of G is a set of independent edges of G covering all vertices of G.Let H be a benzenoid system with a perfect matching,a Clar cover C of H is a spanning subgraph of H each(connected) component of which is either a hexagon or an edge. We dnote the number of hexagons of C by h,(C) andσ(H)=Max[h,(C)|C is a Clar cover of H},then the polynomial P(H,w)=∑sum from i=0 toσ(H)σ(H,i)w~i is called as the Zhang-Zhang polynomial of H,whereσ(H,i) dnotes the number of Clar covers of H having precisely i hexagons and w is an indeterminate or weight associted with hexagons of H.Anti-forcing set S of H is a subset of edges set of H such that the remainder of H obtained by deleting the edges of S that has a unique perfect matching.An Anti-forcing set of the smallest eardinality is called a minimal anti-forcing set and its cardinality is the anti-forcing number of H.In this paper,we give an explicit recurrence expression for the Zhang-Zhang polynomials of the cyclo-polyphenacenes and determine their Clar numbers,numbers of Kekul(?) structures and their first Herndon numbers;And we also give an upper bound of anti-forcing number of the cyclo-polyphenacenes.