On the variable Wiener indices of trees with given maximum degree

The Wiener index of a tree TT obeys the relation W(T)=∑en1(e)⋅n2(e)W(T)=∑en1(e)⋅n2(e), where n1(e)n1(e) and n2(e)n2(e) are the number of vertices adjacent to each of the two end vertices of the edge ee, respectively, and where the summation goes over all edges of TT. Lately, Nikolić, Trinajstić and Randić put forward a novel modification mWmW of the Wiener index, defined as mW(T)=∑e(n1(e)⋅n2(e))−1mW(T)=∑e(n1(e)⋅n2(e))−1. Very recently, Gutman, Vukičević and Z̆erovnik extended the definitions of W(T)W(T) and mW(T)mW(T) to be mWλ(T)=∑e(n1(e)⋅n2(e))λmWλ(T)=∑e(n1(e)⋅n2(e))λ, and they called mWmW the modified Wiener index of TT, and mWλ(T)mWλ(T) the variable Wiener index of TT. Let Δ(T)Δ(T) denote the maximum degree of TT. Let TnTn denote the set of trees on nn vertices, and Tnc={T∈Tn∣Δ(T)=c}. In this paper, we determine the first two largest (resp. smallest) values of mWλ(T)mWλ(T) for λ>0λ>0 (resp. λ<0λ<0) in Tnc, where c≥n2. And we identify the first two largest and first three smallest Wiener indices in Tnc(c≥n2), respectively. Moreover, the first two largest and first two smallest modified Wiener indices in Tnc(c≥n2) are also identified, respectively.