On the normalized distance laplacian eigenvalues of graphs

•The normalized distance Laplacian matrix (DL-matrix) of a connected graph Γ is defined by DL(Γ)=I−Tr(Γ)−1/2=D(Γ)Tr(Γ)−1/2; where D(Γ) is the distance matrix and Tr(Γ) is the diagonal matrix of the vertex transmissions in Γ.•Characterize the graphs having exactly two distinct DL-eigenvalues which in turn solves a conjecture proposed in the literature.•Determine the bounds for the DL-spectral radius and the second smallest eigenvalue of DL(Γ)-matrix, and identify the candidate graphs attaining them.•Identify the classes of graphs whose second smallest DL-eigenvalue is 1 and relate it with the distance spectrum of such graphs.•Introduce the concept of the trace norm (the normalized distance Laplacian energy DLE(Γ) of Γ) of I−DL(Γ).•Obtain some bounds on DLE(Γ) and characterize the corresponding extremal graphs. The normalized distance Laplacian matrix (DL-matrix) of a connected graph Γ is defined by DL(Γ)=I−Tr(Γ)−1/2D(Γ)Tr(Γ)−1/2, where D(Γ) is the distance matrix and Tr(Γ) is the diagonal matrix of the vertex transmissions in Γ. In this article, we present interesting spectral properties of DL(Γ)-matrix. We characterize the graphs having exactly two distinct DL-eigenvalues which in turn solves a conjecture proposed in [26]. We characterize the complete multipartite graphs with three distinct DL-eigenvalues. We present the bounds for the DL-spectral radius and the second smallest eigenvalue of DL(Γ)-matrix and identify the candidate graphs attaining them. We also identify the classes of graphs whose second smallest DL-eigenvalue is 1 and relate it with the distance spectrum of such graphs. Further, we introduce the concept of the trace norm (the normalized distance Laplacian energy DLE(Γ) of Γ) of I−DL(Γ). We obtain some bounds and characterize the corresponding extremal graphs.