Nonexistence of certain Q-polynomial distance-regular graphs

Abstract

I. N. Belousov, A. A. Makhnev, and M. S. Nirova described Q-polynomial distance-regular graphs Γ of diameter 3 for which the graphs Γ2 and Γ3 are strongly regular. Set a = a3. A graph Γ has type (I) if c2 + 1 divides a, type (II) if c2 + 1 divides a + 1, and type (III) if c2 + 1 divides neither a nor a + 1. If Γ is a graph of type (II), then a + 1 = w(c2 + 1), t2 = w(w(c2 + 1) + c2), and either (i) w = s2, t2 = s2(s2(c2 + 1) + c2), (s2(c2 + 1) + c2 is the square of an integer u, c2 = (u2 − s2)/(s2 + 1), t = su, and a = (u2s2 − 1)/(s2 + 1) or (ii) c2 = sw, t2 = w2(sw + 1 + s), sw + 1 + s is the square of an integer u, c2 = (u2 − 1)w/(w + 1), t = uw, a = (u2w2 − 1)/(w + 1), and Γ has intersection array (equation presented) If a graph of type (IIii) is such that w = u, then it has intersection array {w4 + w − 1, w4 − w3, (w2 − w + 1)w; 1, w(w − 1), (w2 − w + 1)w2}. We prove that graphs with such intersection arrays do not exist for even w. © 2019 Krasovskii Institute of Mathematics and Mechanics. All rights reserved.