The Wiener-Hosoya index was firstly introduced by M. Randié¢ in \(2004\). For any tree \(T\), the Wiener-Hosoya index is defined as
\[WH(T)= \sum\limits_{e\in E(T)} (h(e) + h[e])\]
where \(e = uv\) is an arbitrary edge of \(T\), and \(h(e)\) is the product of the numbers of the vertices in each component of \(T – e\), and \(h[e]\) is the product of the numbers of the vertices in each component of \(T- \{u,v\}\). We shall investigate the Wiener-Hosoya index of trees with diameter not larger than \(4\), and characterize the extremal graphs in this paper.
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