Some graft transformations and its application on a distance spectrum

Let D(G)=(di,j)n×n denote the distance matrix of a connected graph G with order n, where dij is equal to the distance between vi and vj 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 Sn′ (obtained from the star Sn 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 Pn′ (obtained from a triangle K3 by attaching pendent path Pn−3 to one of its vertices) has maximal distance spectral radius among unicyclic graphs on n vertices.

► Some new graft transformations that increase distance spectral radius are presented. ► Graphs with minimal distance spectral radius among unicyclic graphs are determined. ► Graphs with maximal distance spectral radius among unicyclic graphs are determined.