Some graft transformations and its application on a distance spectrum

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, we give some graft transformations that decrease and increase ϱ ( G ) and prove that the graph S n ′ (obtained from the star S n on n ( n is not equal to 4, 5) vertices by adding an edge connecting two pendent vertices) has minimal distance spectral radius among unicyclic graphs on n vertices; while P n ′ (obtained from a triangle K 3 by attaching pendent path P n − 3 to one of its vertices) has maximal distance spectral radius among unicyclic graphs on n vertices.