aDepartment of Mathematics, De La Salle University, Manila, Philippines
Abstract:
Let Γ denote a distance-regular graph with diameter D3. Let θ denote a nontrivial eigenvalue of Γ and let denote the corresponding dual eigenvalue sequence. In this paper we prove that Γ is Q-polynomial with respect to θ if and only if the following (i)–(iii) hold:
(i) There exist such that
(1)
(ii) There exist such that the intersection numbers ai satisfy
for 0iD, where and are the scalars which satisfy Eq. (1) for i=0, i=D, respectively.
(iii) for 1iD.
Keywords: Distance-regular graph; Q-polynomial; Association scheme