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.