On Automorphisms of Distance-Regular Graph with Intersection Array $$\{18,15,9;\,1,1,10\}$$ { 18 , 15 , 9 ; 1 , 1 , 10 }

Abstract

Recently, Makhnev and Nirova found intersection arrays of distance-regular graphs with \(\lambda =2\) and at most 4096 vertices. In the case of primitive graphs of diameter 3 with \(\mu = 1\) there corresponding arrays are \(\{18,15,9;1,1,10\}\), \(\{33,30,8;1,1,30\}\) or \(\{39,36,4;1,1,36\}\). In this work, possible orders and subgraphs of fixed points of the hypothetical distance-regular graph with intersection array \(\{18,15,9;1,1,10\}\) are studied. In particular, graph with intersection array \(\{18,15,9;1,1,10\}\) is not vertex symmetric.