The Dual Eigenvalues Of Distance-regular Graphs

Let Γ denote a distance-regular graph of diameter D≥3,with k≥3,a1=0<a2. Assume θ is an eigenvalue of Γ and E is the associated primitive idempotent. The sequence θ0*,θ1*,…,θD*is the dual eigenvalues sequence associated with θ.In this paper, we will study the main properties of the two graphs. The one is the a1=0<a2with classical parameters(D,b,α,β). The other is the regular near polygon. We obtain the relation among the dual eigenvalues.We also study the regular near polygon with a1=1.Main results:●Let Γ be a distance-regular graph of diameter D≥3such that a1=0<a2with classical parameters(D,b,α,β).Assume θ is an eigenvalue of Γ and E is the associated primitine idempotent.The sequence θ0*,θ1*,…,θD*is the dual eigenvalues sequence associated with θ.Let i be an integer with1≤i≤D-1.Then the followings hold:(i)(θi*-θ0*)(θi*-θ2*)=(θ1*-θi+1*)(θ1*-θi-1*).(5)()(θ2*-θi*)(θ0*-θi-1*)=(θ1*-θi-1*)(θ1*-θi*).(6) Moreover,(θ1*-θi+1*)(θ0*-θi-1*)=(θi*-θ1*)(θi*-θ0*).(7)●Let Γ be a distance-regular graph of diameter D≥3with classical parameters (D,b,α,β)such that a1=0<a2.Assume θ is an eigenvalue of Γ and E is the associated primitive idempotent.The sequence θ0*,θ1*,…,θD*is the dual eigenvalues sequence associated with θ.Let i be an integer with1≤i≤D-1.Let Then the followings are equivalent:(ⅰ)<B,B><F,F>-<B,F>...