On the largest eigenvalue of the distance matrix of a connected graph

We report some properties of the largest eigenvalue Λ1 of the distance matrix of a connected graph, in particular, the upper and lower bounds for Λ1 involving the sum of the squares of the distances between all unordered pairs of vertices and the sum of the distances between a given vertex and all other vertices. We also give the relationship between Λ1 and the first Zagreb index and the Wiener index. Additionally, we give the Nordhaus–Gaddum-type result for Λ1.