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 $\rho(G)$ and the proximity $\pi(G)$ 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 $C_4$-free graphs of given order and minimum degree, and we demonstrate that these are close to being best possible.