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The average distance of a vertex $v$ of a connected graph $G$ is thearithmetic mean of the distances from $v$ to all other vertices of $G$. Theproximity $\pi(G)$ and the remoteness $\rho(G)$ of $G$ are the minimum and themaximum of the average distances of the vertices of $G$. In this paper, we giveupper bounds on the difference between the remoteness and proximity, thediameter and proximity, and the radius and proximity of a triangle-free graphwith given order and minimum degree. We derive the latter two results by firstproving lower bounds on the proximity in terms of order, minimum degree andeither diameter or radius. Our bounds are sharp apart from an additiveconstant. We also obtain corresponding bounds for $C_4$-free graphs.

On the difference between proximity and other distance parameters in triangle-free graphs and $C_4$-free graphs