Proof of conjectures on the distance signless Laplacian eigenvalues of graphs

Let G = (V, E) be a simple graph with vertex set V (G) = {v₁, v₂, . . . , v_n} and edge set E(G). The distance signless Laplacian spectral radius of a connected graph G is the spectral radius of the distance signless Laplacian matrix of G, defined as Q(G) = Tr(G) + D(G), where Tr(G) is the diagonal matrix of vertex transmissions of G and D(G) is the distance matrix of G. In [10], Xing et al. determined the graph with minimum distance signless Laplacian spectral radius among the trees with fixed number of vertices. For n ≥ 3, let T_{n-3, 1}¹ be the n-vertex tree of maximum degree n - 2. In this paper, we show that T_{n-3, 1}¹ gives the second minimum distance signless Laplacian spectral radius among the trees with fixed number of vertices. Moreover, we prove two conjectures involving the second largest eigenvalue of the distance signless Laplacian matrix Q(G) of graph G.