Abstract
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 \({\cal G}\) of connected graphs is defined as the sum of the Wiener indices of its members, \(W({\cal G})= \sum_{G \in {\cal
G}} W(G)\). Let \(U_e\) be a unicyclic graph obtained by replacing an edge \(e\) of a tree \(T\) with a fixed length cycle. A simple relation between Wiener indices of the family \(\{ U_e\, | \, e\in E(T) \}\) and a tree \(T\) is presented
for certain positions of the cycle.
