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 arrays {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.