On local structure of a distance-regular graph of Hamming type
Let G be a distance-regular graph with the vertex set V and let \(\partial\) denote the metric on V. For u,v\(\in V\) and non-negative integers r, s, we define \(\Gamma _ r(u)=\{x\in V|\) \(\partial (x,u)=r\}\), \(D^ r_ s(u,v)=\Gamma _ r(u)\cap \Gamma _ s(v)\). Let \(a_ i\), \(b_ i\), \(c_ i\) (0\(\leq i\leq d\), d is the diameter of G) be the usual intersection numbers of G. In the following we assume G is a distance-regular graph with \(c_ 2=2\), \(c_ 3=3.\) Theorem 1. Let u, v be vertices in G with \(\partial (u,v)=3\). Then the number of edges in \(D^ 1_ 2(u,v)\) is at most one. Theorem 2. Let uv be an edge in G and let \(P_ 1,...,P_ m\) be the connected components in \(D^ 1_ 1(u,v)\), \(p_ i=| P_ i|\). Then \[ a_ 2\geq a_ 1-1+\sum ^{m}_{i=1}p_ i(a_ 1-p_ i). \] Theorem 3. If \(a_ 1>2\) ad \(a_ 2=2a_ 1\), \(D^ 1_ 1(u,v)\) is a clique for each edge uv in G.
scientific article; zbMATH DE number 4021194