Maximum spectral radius of graphs with given connectivity, minimum degree and independence number

In this paper we study three families of graphs, one is the graphs of order n   with connectivity κ(G)≤kκ(G)≤k and minimum degree δ(G)≥kδ(G)≥k. We show that among the graphs in this family, the maximum spectral radius is obtained uniquely at Kk+(Kδ−k+1∪Kn−δ−1)Kk+(Kδ−k+1∪Kn−δ−1). Another family of the graphs we study is the family of bipartite graphs with order n and connectivity k  . We show that among the graphs in this family the maximum spectral radius is obtained at a graph modified from K⌊n/2⌋,n−1−⌊n/2⌋K⌊n/2⌋,n−1−⌊n/2⌋. The third family of graphs we have studied is the family of graphs with order n, connectivity k and independence number r. We determine the graphs in this family that have the maximum spectral radius.