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

Let D ( G ) = ( d i , j ) n × n denote the distance matrix of a connected graph G with order n , where d i j is equal to the distance between v i and v j in G . The largest eigenvalue of D ( G ) is called the distance spectral radius of graph G , denoted by ϱ ( G ) . In this paper, some graft transformations that decrease or increase ϱ ( G ) are given. With them, for the graphs with both order n and k 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 − 2 ; for k = 1 , 2 , 3 , n − 1 , the extremal graphs with the maximum distance spectral radius are completely characterized.