Обратные задачи в классе Q-полиномиальных графов

Abstract

In the class of distance-regular graphs Γ of diameter 3 with a pseudogeometric graph Γ3, feasible intersection arrays for the partial geometry were found for networks by Makhnev, Golubyatnikov, and Guo; for dual networks by Belousov and Makhnev; and for generalized quadrangles by Makhnev and Nirova. These authors obtained four infinite series of feasible intersection arrays of distance-regular graphs: {c2(u2 − m2) + 2c2m − c2 − 1, c2(u2 − m2), (c2 − 1)(u2 − m2) + 2c2m − c2; 1, c2, u2 − m2}, {mt, (t + 1)(m − 1), t + 1; 1, 1, (m − 1)t} for m ≤ t, {lt, (t − 1)(l − 1), t + 1; 1, t − 1, (l − 1)t}, and {a(p + 1), ap, a + 1; 1, a, ap}. We find all feasible intersection arrays of Q-polynomial graphs from these series. In particular, we show that, among these infinite families of feasible arrays, only two arrays ({7, 6, 5; 1, 2, 3} (folded 7-cube) and {191, 156, 153; 1, 4, 39}) correspond to Q-polynomial graphs. © 2020 Sverre Raffnsoe. All rights reserved.This work was supported by the Russian Foundation for Basic Research – the National Natural Science Foundation of China (project no. 20-51-53013_a)