Wiener Polynomials for Multi-Rings Paraffin Structures

The distance between any two vertices u and v in a connected graph G is defined as the length of the shortest path between them, and it is denoted by d(u,v).The sum of distances for all unordered pairs of distinct vertices in G represents Wiener index.
The number of pairs of vertices G which are distance k apart is denoted by d(G,k), it is clear that the number of d(G,k) is graphical invariant, and the Wiener polynomial of graph G is a generating function of the sequence d(G,k).
In this paper, we find the Wiener polynomial of multi-circles of paraffin structural, and this formula which we obtained is better than the formula prove in [5] , because we are able to evaluate coefficients for any limited power of x without depending on the number of circles , and we find the Wiener index and average distance for this structural.
Lastly, we contracted a MATLAB program to evaluate the Wiener polynomial coefficient ,Wiener index and average distance.