Geodetic graphs of diameter two and some related structures

AbstractWe show a connection between 2-connected geodetic graphs of diameter two and certain geometric structures; we apply this result, proving that these graphs fall into three classes: (1) strongly regular graphs with μ = 1; (2) pyramids; (3) π-graphs. (The definitions for (1), (2), and (3) are given in the text.) We also give some results on graphs of type (3), including some non-existence conditions. Finally, we consider a more general class of graphs, showing a connection between some of them and Sperner spaces