Characterizing Bipartite Distance-Regularized Graphs with Vertices of Eccentricity 4

Consider a bipartite distance-regularized graph Γ Γ with color partitions Y and $$Y'$$ Y ′ . Notably, all vertices in partition Y (and similarly in $$Y'$$ Y ′ ) exhibit a shared eccentricity denoted as D (and $$D'$$ D ′ , respectively). The characterization of bipartite distance-regularized graphs, specifically those with D ≤ 3 D ≤ 3 , in relation to the incidence structures they represent is well established. However, when $$D=4$$ D = 4 , there are only two possible scenarios: either $$D'=3$$ D ′ = 3 or $$D'=4$$ D ′ = 4 . The instance where $$D=4$$ D = 4 and $$D'=3$$ D ′ = 3 has been previously investigated. In this paper, we establish a one-to-one correspondence between the incidence graphs of quasi-symmetric SPBIBDs with parameters (v, b, r, k, λ ₁, 0) ( v , b , r , k , λ 1 , 0 ) of type $$(k-1, t)$$ ( k - 1 , t ) , featuring intersection numbers $$x=0$$ x = 0 and $$y>0$$ y > 0 (where y ≤ t < k y ≤ t < k ), and bipartite distance-regularized graphs with $$D=D'=4$$ D = D ′ = 4 . Moreover, our investigations result in the systematic classification of 2- Y -homogeneous bipartite distance-regularized graphs, which are incidence graphs of quasi-symmetric SPBIBDs with parameters (v,b,r,k, λ ₁,0) ( v , b , r , k , λ 1 , 0 ) of type $$(k-1,t)$$ ( k - 1 , t ) with intersection numbers $$x=0$$ x = 0 and $$y=1$$ y = 1 .