Regularity and transitivity in graphs

Graphs with high regularity and transitivity conditions are studied. The first graphs considered are graphs where each vertex has an intersection array (possibly differing from that of other vertices). These graphs are called distance-regularised and are shown to be distance-regular or bipartite with each bipartition having the same intersection array. The latter graphs are called distance-biregular. This leads to the study of distance-biregular graphs. The derived graphs of a distance-biregular graph are shown to be distance-regular and the notion of feasibility for a distance-regular graph is extended to the biregular case. The study of the intersection arrays of distance-biregular graphs is concluded with a bound on the diameter in terms of the girth and valencies. Special classes of distance-biregular graphs are also studied. Distance-biregular graphs with 2-valent vertices are shown to be the subdivision graphs of cages …