Some graft transformations and its applications on the distance spectral radius of a graph

Let D(G)=(di,j)n×nD(G)=(di,j)n×n denote the distance matrix of a connected graph GG with order nn, where dijdij is equal to the distance between vivi and vjvj in GG. The largest eigenvalue of D(G)D(G) is called the distance spectral radius of graph GG, denoted by ϱ(G)ϱ(G). In this paper, some graft transformations that decrease or increase ϱ(G)ϱ(G) are given. With them, for the graphs with both order nn and kk pendant vertices, the extremal graphs with the minimum distance spectral radius are completely characterized; the extremal graph with the maximum distance spectral radius is shown to be a dumbbell graph (obtained by attaching some pendant edges to each pendant vertex of a path respectively) when 2≤k≤n−22≤k≤n−2; for k=1,2,3,n−1k=1,2,3,n−1, the extremal graphs with the maximum distance spectral radius are completely characterized.