On the Wiener index of two families generated by joining a graph to a tree

The Wiener index W(G) of a graph G is the sum of distances between all vertices of G. The Wiener index of a family of connected graphs is defined as the sum of the Wiener indices of its members. Two families of graphs can be constructed by identifying a fixed vertex of an arbitrary graph F with vertices or subdivision vertices of an arbitrary tree T of order n. Let G(v) be a graph obtained by identifying a fixed vertex of F with a vertex v of T. The first family V = {G(v)vertical bar v is an element of V (T)} contains n graphs. Denote by G(ve) a graph obtained by identifying the same fixed vertex of F with the subdivision vertex v(e) of an edge e in T. The second family epsilon = {G(ve) vertical bar e is an element of E(T)} contains n - 1 graphs. It is proved that W(V) = W (epsilon) if and only if W (F) = 2W (T).