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 Gv be a graph obtained by identifying a fixed vertex of F with a vertex v of T . The first family V = {Gv | v ∈ V (T )} contains n graphs. Denote by Gve a graph obtained by identifying the same fixed vertex of F with the subdivision vertex ve of an edge e in T . The second family E = {Gve | e ∈ E(T )} contains n − 1 graphs. It is proved that W (V) = W (E) if and only if W (F ) = 2W (T ).