Let G denote a distance-biregular graph with bipartite parts Y and Y’. Let D denote the eccentricity of vertices in Y. Given a vertex z, let \Gamma_i(z) denote the set of all vertices which are at distance i from z. For vertices x and y, let \Gamma_{i,j}(x,y) denote the collection of all vertices which are at distance i from x and at distance j from y.
In this talk, we will show necessary and sufficient conditions on the intersection array of G for which the given graph has one of the following two combinatorial structures:
for all i (1 \leq i \leq D-2) and for all x\in Y, y\in \G_2(x) and z \in \G_{i,i}(x,y) the number of vertices in \G_{1,1}(x,y) which are at distance i-1 from z is independent of the choice of x,y and z.
for all i (1 \leq i \leq D-1) and for all x\in Y, y\in \G_2(x) and z \in \G_{i,i}(x,y) the number of vertices in \G_{1,1}(x,y) which are at distance i-1 from z is independent of the choice of x,y and z.
Distance-biregular graphs with the previous combinatorial structures are called almost 2-Y-homogeneous and 2-Y-homogeneous, respectively. Several examples will also be presented. This is joint work with Safet Penjić.