On the eigenvalues of the distance matrix of graphs with given number of pendant vertices

Let G be a simple connected graph with vertices v₁, v₂ …, v_n. The distance matrix of G, denoted by D(G), is the n×n matrix whose (i, j)^th element is equal to d(v_i, v_j) (the length of a shortest path between v_i and v_j). Let ℙ(n, r) be the family of all connected graphs of order n having r pendant vertices. In this paper, we obtain the distance spectrum of various subfamilies of ℙ(n, r), like pineapple graphs, kite graphs, double star graphs, etc. We also determine the graphs with the largest and smallest spectral radii belonging to these families. Finally, we give a lower bound for the smallest distance eigenvalue of certain kite graphs in terms of minimum transmission.