Analytical expressions for Wiener indices of n-circumscribed peri-condensed benzenoid graphs

We have employed cut axes and vertex partitioning (I-partition) methods to obtain exact analytical expressions for Wiener indices of n-circumscribed peri-condensed benezenoid graphs as a function of n, the order of circumscribing. Such expressions have not been obtained before for nth order circumscribing of peri-condensed benzenoid graphs with the exception of n-Circum-coronene which becomes a honeycomb lattice as n goes to infinity. The Wiener indices of these n-circum peri-condensed benzenoids are found to be polynomials of degree 5. Such expressions can be critical in topological characterization of peri-condensed benzenoid sheets of any order of circumscribing. A number of examples are provided ranging from Circum-polyacenes, Circum-ovalenes, Circum-pyrenes to more complex Circum-peri-condensed graphs of wide ranging complexities.