A characterization of Q-polynomial distance-regular graphs Original Research Article

Let image denote a distance-regular graph with diameter image. Let image denote a nontrivial eigenvalue of image and let image denote the corresponding dual eigenvalue sequence. In this paper we prove that image is Q-polynomial with respect to image if and only if the following (i)–(iii) hold: (i) There exist image such that equation(1) (ii) There exist image such that the intersection numbers image satisfy for image, where image and image are the scalars which satisfy Eq. for image, image, respectively. (iii) image for image.