On the variable Wiener indices of trees with given maximum degree

The Wiener index of a tree T obeys the relation W ( T ) = ∑ e n 1 ( e ) ⋅ 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 m W of the Wiener index, defined as m W ( T ) = ∑ e ( n 1 ( e ) ⋅ n 2 ( e ) ) − 1 . Very recently, Gutman, Vuki c ̌ ević and Z ̆ erovnik extended the definitions of W ( T ) and m W ( T ) to be m W λ ( T ) = ∑ e ( n 1 ( e ) ⋅ n 2 ( e ) ) λ , and they called m W the modified Wiener index of T , and m W λ ( T ) the variable Wiener index of T . Let Δ ( T ) denote the maximum degree of T . Let T n denote the set of trees on n vertices, and T n c = { T ∈ T n ∣ Δ ( T ) = c } . In this paper, we determine the first two largest (resp. smallest) values of m W λ ( T ) for λ > 0 (resp. λ < 0 ) in T n c , where c ≥ n 2 . And we identify the first two largest and first three smallest Wiener indices in T n c ( c ≥ n 2 ) , respectively. Moreover, the first two largest and first two smallest modified Wiener indices in T n c ( c ≥ n 2 ) are also identified, respectively.