Characterizing Bipartite Distance-Regularized Graphs with Vertices of Eccentricity 4

Consider a bipartite distance-regularized graph \(\Gamma \) with color partitions Y and \(Y'\). Notably, all vertices in partition Y (and similarly in \(Y'\)) exhibit a shared eccentricity denoted as D (and \(D'\), respectively). The characterization of bipartite distance-regularized graphs, specifically those with \(D \le 3\), in relation to the incidence structures they represent is well established. However, when \(D=4\), there are only two possible scenarios: either \(D'=3\) or \(D'=4\). The instance where \(D=4\) and \(D'=3\) has been previously investigated. In this paper, we establish a one-to-one correspondence between the incidence graphs of quasi-symmetric SPBIBDs with parameters \((v, b, r, k, \lambda _1, 0)\) of type \((k-1, t)\), featuring intersection numbers \(x=0\) and \(y>0\) (where \(y \le t < k\)), and bipartite distance-regularized graphs with \(D=D'=4\). 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 \((v,b,r,k, \lambda _1,0)\) of type \((k-1,t)\) with intersection numbers \(x=0\) and \(y=1\).