Distance-regular graph with intersection array {27, 20, 7; 1, 4, 21} does not exist

Abstract

In the class of distance-regular graphs of diameter 3 there are 5 intersection arrays of graphs with at most 28 vertices and noninteger eigenvalue. These arrays are {18, 14, 5; 1, 2, 14}, {18, 15, 9; 1, 1, 10}, {21, 16, 10; 1, 2, 12}, {24, 21, 3; 1, 3, 18}, and {27, 20, 7; 1, 4, 21}. Automorphisms of graphs with intersection ar- rays {18, 15, 9; 1, 1, 10} and {24, 21, 3; 1, 3, 18} were found earlier by A.A. Makhnev and D.V. Paduchikh. In this paper, it is proved that a graph with the intersection array {27, 20, 7; 1, 4, 21} does not exist. © 2020, Krasovskii Institute of Mathematics and Mechanics. All rights reserved.