Trees preserving Wiener index in two classes of graphs

The existence problem on trees preserving the Wiener index of two classes of graphs is studied in this paper. The Wiener index W(G) of a connected graph G is the sum of distances among all pairs of vertices in G. If there is a connected subtree T of a given connected graph G such that W(G)=W(T), then T is called a preserving the Wiener index tree of G. In this paper, the graph S(s,t,l,k,s,t,l) is defined as a multi-fan graph with pendent edges and the graph G(s,t,l,m,k) is defined as a group of multifan graphs with pendent edges. By using the definition and the properties of Wiener index of a graph, it is proved that there exist subtrees preserving Wiener index in those two classes of graphs.