Characterizing Bipartite Distance-Regularized Graphs with Vertices of Eccentricity 4

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