On the second largest eigenvalue of the generalized distance matrix of graphs

For a simple connected graph G, let D(G), Tr(G), DL(G) and DQ(G), respectively be the distance matrix, the diagonal matrix of the vertex transmissions, distance Laplacian matrix and the distance signless Laplacian matrix of a graph G. The generalized distance matrix Dα(G) is a convex linear combinations of Tr(G) and D(G) and defined as Dα(G)=αTr(G)+(1−α)D(G), 0≤α≤1. As D0(G)=D(G), 2D12(G)=DQ(G), D1(G)=Tr(G) and Dα(G)−Dβ(G)=(α−β)DL(G), this matrix reduces to merging the distance spectral, distance Laplacian spectral and distance signless Laplacian spectral theories. In this paper, we take effort to obtain some upper and lower bounds for the second largest eigenvalue of the generalized distance matrix of graphs, in terms of various graph parameters. The graphs attaining the corresponding bounds are characterized. As application, we give a confirmation to a conjecture about the second largest distance signless Laplacian eigenvalue of a connected graph due to Aouchiche and Hansen [6]. We also show that the star graph Sn has the smallest second largest generalized distance eigenvalue among all trees of order n. As application, we give a confirmation to a conjecture about the second largest distance signless Laplacian eigenvalue of a tree due to Aouchiche and Hansen [10].