Proximity and remoteness in triangle-free and C₄-free graphs in terms of order and minimum degree

Let G be a finite, connected graph. The average distance of a vertex v of G is the arithmetic mean of the distances from v to all other vertices of G. The remoteness ρ(G) and the proximity π(G) of G are the maximum and the minimum of the average distances of the vertices of G. In this paper, we present a sharp upper bound on the remoteness of a triangle-free graph of given order and minimum degree, and a corresponding bound on the proximity, which is sharp apart from an additive constant. We also present upper bounds on the remoteness and proximity of C₄-free graphs of given order and minimum degree, and we demonstrate that these are close to being best possible.