Maximum Zagreb index, minimum hyper-Wiener index and graph connectivity

In this work we show that among all nn-vertex graphs with edge or vertex connectivity kk, the graph G=Kk∨(K1+Kn−k−1)G=Kk∨(K1+Kn−k−1), the join of KkKk, the complete graph on kk vertices, with the disjoint union of K1K1 and Kn−k−1Kn−k−1, is the unique graph with maximum sum of squares of vertex degrees. This graph is also the unique nn-vertex graph with edge or vertex connectivity kk whose hyper-Wiener index is minimum.