Ordering trees by their distance spectral radii

Let D(G)D(G) be the distance matrix of a connected graph GG. The distance spectral radius of GG, denoted by ∂1(G)∂1(G), is the largest eigenvalue of D(G)D(G). In this paper we present a new transformation of a certain graph GG that decreases ∂1(G)∂1(G). With the transformation, we partially confirm a conjecture proposed by Stevanović and Ilić [17] by showing that, if Δ≥⌈n2⌉, the double star SΔ,n−ΔSΔ,n−Δ uniquely minimizes the distance spectral radius among all trees on nn vertices with maximum degree ΔΔ. Moreover, the trees on n≥10n≥10 vertices with the fourth and fifth least distance spectral radii are characterized.