Maximum/minimum orders of graphs with given degree and given diameter and/or girth

The well-known Moore graphs (including odd-length cycles, the complete graphs, the Petersen graph and the Hoffman-Singleton graph) are regular graphs of maximum conceivable order with given degree and diameter, or equivalently, regular graphs of minimum conceivable order with given degree and girth (sometimes called cages), according to the well-known Moore bound.  

More generally, the task of finding the largest regular graph with given degree and diameter is called the degree-diameter problem, and the corresponding one for given degree and girth is called the cage problem. Various people in the combinatorial community have contributed answers or partial answers to these problems.

At a BIRS workshop at Banff in May 2023, some investigations were made into the stronger notion of a degree-diameter-girth problem, namely finding the largest regular graph with given degree, diameter and girth.  I will report on some of what was discovered, after reviewing some of the background to the degree-diameter and cage problems.