Characterizing bipartite distance-regularized graphs with vertices of eccentricity 4

TL;DR

This paper establishes a precise correspondence between bipartite distance-regularized graphs with eccentricity 4 and certain incidence graphs of quasi-symmetric SPBIBDs, advancing the structural understanding of these graphs.

Contribution

It proves a one-to-one correspondence between bipartite distance-regularized graphs with eccentricity 4 and incidence graphs of specific quasi-symmetric SPBIBDs, extending existing characterizations.

Findings

Established a correspondence between bipartite distance-regularized graphs and SPBIBDs.

Characterized graphs with eccentricity 4 in terms of incidence structures.

Extended the understanding of graph-incidence structure relationships.

Abstract

The characterization of bipartite distance-regularized graphs, where some vertices have eccentricity less than four, in terms of the incidence structures of which they are incidence graphs, is known. In this paper we prove that there is a one-to-one correspondence between the incidence graphs of quasi-symmetric SPBIBDs with parameters $(v,b,r,k, \lambda_1,0)$ of type $(k-1,t)$ with intersection numbers $x=0$ and $y>0$, where $0< y\leq t<k$ , and bipartite distance-regularized graphs with $D=D'=4$.