On distance spectral radius of graphs with given number of pendant paths of fixed length

A pendant path of length r with r >= 1 in a graph G is a path u(0)u(1) ... u(r) in G such that the degree of u(0) is one, the degree of u(r) is at least two, and if r >= 2, then the degree of u(i) is two for any i with 1 <= i <= r - 1. The distance spectral radius of a connected graph is the largest eigenvalue of the distance matrix of the graph. We determine the unique tree that maximizes (respectively minimizes) the distance spectral radius over all trees on n vertices with k pendant paths of length r, and also determine the unique graph that minimizes the distance spectral radius over all connected graphs on n vertices with k pendant paths of length r, where k, r >= 1 and kr < n - 1.