2-Y-homogeneous distance-biregular graphs

Distance-regular graphs are a central and comparatively well-understood class in algebraic combinatorics, with a well-controlled distance structure around every vertex. Their bipartite analogues, distance-biregular graphs, also have a well-controlled distance structure, but its parameters may depend on which part of the bipartition a vertex belongs to. This makes them a natural setting in which to ask how far ideas from distance-regular graph theory can be extended. In this talk, we introduce 2-Y-homogeneity, a one-sided analogue of 2-homogeneity, and use it to investigate the structure of distance-biregular graphs that satisfy this property.

The classification results obtained so far reveal links with combinatorial designs. At diameter three, 2-Y-homogeneity is connected to quasi-symmetric 3-designs; at diameter four, to incidence structures arising from quasi-symmetric 1-designs. At diameter five, we prove that no such graphs exist when certain scalar c’_2 ≥ 3. I will conclude with the remaining open cases and conjectures,  and share why completing this classification remains a long-standing goal of my research.