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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 $\rho(G)$ and the proximity $\pi(G)$ of $G$ are the maximumand 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 ofgiven order and minimum degree, and a corresponding bound on the proximity,which is sharp apart from an additive constant. We also present upper bounds onthe remoteness and proximity of $C_4$-free graphs of given order and minimumdegree, and we demonstrate that these are close to being best possible.

Proximity and remoteness in triangle-free and C_4-free graphs in terms of order and minimum degree