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

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