The maximum girth and minimum circumference of graphs with prescribed radius and diameter

Ostrand posed the following two questions in 1973. (1) What is the maximum girth of a graph with radius r and diameter d? (2) What is the minimum circumference of a graph with radius r and diameter d? Question 2 has been answered by Hrnčiar who proves that if d≤2r−2 the minimum circumference is 4r−2d. In this note we first answer Question 1 by proving that the maximum girth is 2r+1. This improves on the obvious upper bound 2d+1 and implies that every Moore graph is self-centered. We then prove a property of the blocks of a graph which implies Hrnčiar's result.