The Use Of Distance Regular Graphs Of Sub-space Structure For Distance Biregular Graph,

LetΓbe a d-bounded distance-regular graph with diameter d≥3 and with geo-metric parameters(d,b,α)．Pick x∈V(Γ)and let P(x)be the set of subspaces contain-ing x．Suppose P(x,m)and P(x,m+1)are two sets consisting of subspaces in P(x)with diameter m and m+1,respectively,where 1≤m1∈Q,△₂∈L,△₁ is adjacent to△₂ if and only if△₁(?)△₂．We prove that (?)is a distance-biregular graph with diameter max{2min{m,d-m},2min{m+1,d-m-1},2min{m,d-m-1}+1}and compute its intersection numbers．In particular,if thediameter ofΓis d=2m+1≥3,then (?) is a distance-regular graph with diameter 2m+1,its intersection numbers are also computed．The following is our main results:Theorem 3．1 Let (?) be the graph constructed above．Then the following(i)-(iii)hold．(i)For any△₁,△₂∈Q,(?)^*(△₁,△₂)=2i if and only if d(△₁∩△₂)=m-i,where 0≤i≤min{m,d-m}；(ii)For any△₁,△₂∈L,(?)^*(△₁,△₂)=2i if and only if d(△₁∩△₂)=m+1-i,where 0≤i≤min{m+1,d-m-1}；(iii)For any△₁∈Q,△₂∈L,(?)^*(△₁,△₂)=2i+1 if and only if d(△₁∩△₂)=m-i,where 0≤i≤min{m,d-m-1}；Theorem 3．2 (?) is a distance-biregular graph with diameter max{2min{m,d-m},2min{m+1,d-m-1},2min{m,d-m-1}+1}and intersection numbers are where (?)_b² are Gaussian binomial coefficients with basis b²．Theorem 3．3 If d=2m+1≥3,then (?) is a distance-regular graph with diameter2m+1 and intersection numbers arewhere (?)_b² are Gaussian binomial coefficients with basis b²．...