Bounding the $k$-Steiner Wiener and Wiener-Type Indices of Trees in Terms of Eccentric Sequence

The eccentric sequence of a connected graph \(G\) is the nondecreasing eccentricities its vertices. Wiener index sum distances between all unordered pairs vertices \(G\). unique trees that minimise among with given were recently determined by present authors. In this paper we show these results hold not only for index, but large class distance-based topological indices which term Wiener-type indices. Particular cases include hyper-Wiener Harary generalised \(W^{\lambda }\) \(\lambda >0\) and <0\), reciprocal complementary index. Our imply unify known bounds on order diameter.We also similar \(k\)-Steiner sequence. Steiner distance set \(A\subseteq V(G)\) minimum number edges in subtree whose vertex contains \(A\), \(k\)-element subsets \(V(G)\). As corollary, obtain sharp lower bound diameter, determine extremal tree unique, thereby correcting an error literature.