Distance biregular bipartite graphs
A bipartite connected graph \(B\) is called distance biregular if for every pair of vertices \(x,y\) the number of neighbours \(z\) of \(y\) such that \(d(x,z) = d(x,y) - 1\) depends only on the distance \(d(x,y)\) and the stable components containing \(x\) and \(y\). In particular all vertices of the stable component \(G\) resp. \(G'\) will have the same degree \(b_ 0\) resp. \(b_ 0'\), if \(b_ 0 = b_ 1\) then this definition agrees indeed with the definition of a bipartite distance regular graph. Examples of such graphs can be found easily by taking the incidence graph of geometries whose point graph as well as its block graph is distance regular. For instance the incidence graph of a partial geometry is a distance biregular graph. Note that geometries that yield (bipartite) distance biregular incidence graphs are known as regular \((g,d_ p,d_ l)\)-gons where \(2g\) is the girth of the incidence graph and \(d_ p\) (resp. \(d_ l)\) is the point-diameter (resp. line-diameter) of the graph, i.e. it is the maximum value taken by the distance function \(d(x)\) which counts the length of the path starting at a point (resp. a line) \(x\). (See [\textit{F. Buekenhout}, Finite geometries, Proc. Conf. Hon. T. G. Ostrom, Wash. State Univ. 1981, Lect. Notes Pure Appl. Math. 82, 93-111 (1983; Zbl 0526.51008)] and [\textit{F. Buekenhout} and \textit{H. Van Maldeghem}, A characterization of some rank 2 incidence geometries by their automorphism group, Mitt. Math. Sem. Gießen 218, 1-70 (1994)].) In this paper the author gives an overview of some graph theoretical results, especially he extends the known techniques (such as the adjacency algebra, the Hecke algebra, the Krein inequalities, ...) for distance regular graphs to the distance biregular graphs. Note that each bipartite distance biregular graph \(B\) yields indeed two distance regular graphs with vertex set \(G\), resp. \(G'\) where the adjacency is defined by having distance 2 in \(B\). However the Krein conditions on \(B\) are not consequences of the Krein conditions on \(G\) and \(G'\) as shown through an example by the author.
scientific article; zbMATH DE number 597747