Geodetic graphs of diameter two and some related structures (Q1065024)

It is shown a connection between 2-connected geodetic graphs of diameter two and certain geometric structures; applying this result, it is shown that these graphs fall into three classes: (1) strongly regular graphs with \(\mu =1\); (2) pyramids; (3) \(\pi\)-graphs. (The definitions for (1), (2) and (3) are in the text.) We give also some results on graphs of type (3), including some non-existence conditions. In the last section it is considered a more general class of graphs, showing a connection between some of them and Sperner spaces.