Preserving Wiener index trees in two classes of join graphs

The Wiener index W(G) of a connected graph G is the sum of distances among all pairs of vertices of G. Given a connected graph G, if there is a connected subtree T of G such that W(G) =W(T), then we call T is a preserving the Wiener index tree of G. By using graph theory Wiener index calculation method and diophantine equation method, an infinite number of subtrees which preserving the Wiener index in two classes of join graphs with some specific conditions are proved.