Some Shilla Graphs with b = 5 do not Exist

Abstract

A Shilla graph is a distance-regular graph of diameter 3 that has a second eigenvalue equal to a = a3 . Koolen and Park found admissible arrays of intersections of the Shill graphs with b = 3 (there were 12 of them). Belousov I.N. found feasible intersection arrays of the Shilla graphs with b = 4 (there were 50 of them) and b = 5 (there were 82 of them). It is proved in the paper that distance-regular Schill graphs with b = 5 and intersection arrays {305,248,62;1,2,244}, {315,256,64;1,2,252}, {345,280,64;1,4,276},{615,496,124;1,4,492}, {815,656,164;1,2,652}, {855,688,172;1,4,684}, {855,688,170;1,5,684}, {910,732,180;1,10, 728}, {1000,804,201;1,3,800}, {1045,840,210;1,6,836}, {1055,848,212;1,4,844}, {1080,868,215;1,5,864}, {1155,928,232;1,2,924}, {1185,952,245;1,5,948}, {1235,992,248;1,8,988},{1535,1232,308;1,8,1228}, {1560,1252,310;1,10,1248}, {1615,1296,324;1,12,1292}, {1665,1336,334;1,2,1332} do not exist.