Special Regular Graphs

A connected graph Г is called distance-regular if there are integers b _i , c _i (i ≥ 0) such that for any two points γ, δ ε Г at distance i = d(γ,δ), there are precisely c _i neighbours of δ in Г_i−1(γ) and b _i neighbours of δ in Г_i+1(γ). In particular, Г is regular of valency k = b₀. The sequence $$\iota (\Gamma ): = \{ {{b}_{0}},{{b}_{1}}, \cdots ,{{b}_{{d - 1}}};{{c}_{1}},{{c}_{2}}, \cdots ,{{c}_{d}}\} ,$$where d is the diameter of Г, is called the intersection array of Г (cf. Biggs [71]); the numbers c _i , b _i , and a _i , where1$${{a}_{i}} = k - {{b}_{i}} - {{c}_{i}}\;(i = 0, \ldots ,d) $$is the number of neighbours of δ in Γ _i (γ) for d(γ,δ) = i, are called the intersection numbers of Γ. Clearly 2$$\begin{array}{*{20}{c}} {{{b}_{0}} = k,} & {{{b}_{d}} = {{c}_{0}} = 0,} & {{{c}_{1}} = 1.} \\ \end{array}$$