Proximity, remoteness and girth in graphs

The proximity π of a graph G is the minimum average distance from a vertex of G to all others. Similarly, the remoteness of G is the maximum average distance from a vertex to all others. The girth g of a graph G is the length of its smallest cycle. In this paper, we provide and prove sharp lower and upper bounds, in terms of the order n of G, on the difference, the sum, the ratio and the product of the proximity and the girth. We do the same for the remoteness and the girth, except for the lower bound on ρ/g, which is already known.