On the normalized distance laplacian eigenvalues of graphs

The normalized distance Laplacian matrix (Dc-matrix) of a connected graph Gamma is defined by Dc(Gamma)= I- T r(Gamma)-1 / 2D(Gamma)T r(Gamma)-1 / 2, where D(Gamma) is the distance matrix and T r(Gamma) is the diagonal matrix of the vertex transmissions in Gamma. In this article, we present interest-ing spectral properties of Dc(Gamma)-matrix. We characterize the graphs having exactly two distinct Dc-eigenvalues which in turn solves a conjecture proposed in [26]. We charac-terize the complete multipartite graphs with three distinct Dc-eigenvalues. We present the bounds for the Dc-spectral radius and the second smallest eigenvalue of Dc(Gamma)-matrix and identify the candidate graphs attaining them. We also identify the classes of graphs whose second smallest Dc-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 DcE(Gamma) of Gamma) of I- Dc(Gamma). We obtain some bounds and characterize the corre-sponding extremal graphs.(c) 2022 Elsevier Inc. All rights reserved.