On the variable Wiener indices of trees with given maximum degree. (English) Zbl 1205.05073
Summary: The Wiener index of a tree \(T\) obeys the relation \(W(T)=\sum_en_{1}(e)\cdot n_{2}(e)\), where \(n_{1}(e)\) and \(n_{2}(e)\) are the number of vertices adjacent to each of the two end vertices of the edge \(e\), respectively, and where the summation goes over all edges of \(T\). Lately, Nikolić, Trinajstić and Randić put forward a novel modification \(^mW\) of the Wiener index, defined as \(^mW(T)=\sum e(n_{1}(e)\cdot n_{2}(e))^{ - 1}\). Very recently, Gutman, Vukičević and Žerovnik extended the definitions of \(W(T)\) and \(^mW(T)\) to be \(^mW\lambda (T)=\sum e(n_{1}(e)\cdot n_{2}(e))\lambda \), and they called \(^mW\) the modified Wiener index of \(T\), and \(^mW\lambda (T)\) the variable Wiener index of \(T\). Let \(\Delta (T)\) denote the maximum degree of \(T\). Let \(\mathcal T_n\) denote the set of trees on \(n\) vertices, and \(\mathcal T_n^c=\{T\in\mathcal T_n| \Delta(T)=c\}\). In this paper, we determine the first two largest (resp. smallest) values of \(mW\lambda (T)\) for \(\lambda >0\) (resp. \(\lambda <0\)) in \(\mathcal T^c_n\), where \(c\geq\frac12\). And we identify the first two largest and first three smallest Wiener indices in \(\mathcal T^c_n(c\geq\frac12)\), respectively. Moreover, the first two largest and first two smallest modified Wiener indices in \(\mathcal T^c_n(c\geq\frac12)\) are also identified, respectively.
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