
Let G be a connected graph. For a fixed edge uv of G, put \(D^ i_ j=D^ i_ j(u,v)=\Gamma_ i(u)\cap \Gamma_ j(v).\) The family \(\{D^ i_ j\}_{i,j}\) is called the intersection diagram of G with respect to the edge uv. We have obtained several applications of intersection diagrams to distance-regular graphs. In this paper we apply the intersection diagrams to distance-biregular graphs to obtain the following results. Theorem 2. Let G be a distance-biregulr graph with girth \(g\equiv 0\) (mod 4). Let \(s+1\) and \(t+1\) be the valency of G and assume s and t are relatively prime. Then G is isomorphic to a generalized polygon. Theorem 3. Let G be a distance-biregular graph of girth \(g\equiv 0\) (mod 4). Assume G has the valency \(s+1\) and 3. Then one of the following two cases occurs: (1) G is a generalized polygon; (2) \(s=2^ h-2\) for some h with \(h\leq g/2\), \(d(G)\leq g/2+g/h\).
Computational Theory and Mathematics, generalized polygon, Association schemes, strongly regular graphs, Discrete Mathematics and Combinatorics, intersection diagram, distance-biregulr graph, Theoretical Computer Science
Computational Theory and Mathematics, generalized polygon, Association schemes, strongly regular graphs, Discrete Mathematics and Combinatorics, intersection diagram, distance-biregulr graph, Theoretical Computer Science
| selected citations These citations are derived from selected sources. This is an alternative to the "Influence" indicator, which also reflects the overall/total impact of an article in the research community at large, based on the underlying citation network (diachronically). | 7 | |
| popularity This indicator reflects the "current" impact/attention (the "hype") of an article in the research community at large, based on the underlying citation network. | Average | |
| influence This indicator reflects the overall/total impact of an article in the research community at large, based on the underlying citation network (diachronically). | Top 10% | |
| impulse This indicator reflects the initial momentum of an article directly after its publication, based on the underlying citation network. | Average |
