Extremal values for the spectral radius of the normalized distance Laplacian

The normalized distance Laplacian of a graph $G$ is defined as ${\mathcal{D}}^{\mathcal{L}}(G)=T(G{)}^{-1\mathrm{/}2}(T(G)-\mathcal{D}(G))T(G{)}^{-1\mathrm{/}2}$ where $\mathcal{D}(G)$ is the matrix with pairwise distances between vertices and $T(G)$ is the diagonal transmission matrix. In this project, we study the minimum and maximum spectral radii associated with this matrix, and the structures of the graphs that achieve these values. In particular, we prove a conjecture of Reinhart that the complete graph is the unique graph with minimum spectral radius, and we give several partial results towards a second conjecture of Reinhart regarding which graph has the maximum spectral radius.