The prime spectrum of an automorphism group of an 𝐴𝑇4(𝑝,𝑝+2,𝑟)-graph

Abstract

The present paper is devoted classification of A T 4 ( p , p + 2 , r ) \mathrm {AT4}(p,p+2,r) -graphs. There is a unique A T 4 ( p , p + 2 , r ) \mathrm {AT4}(p,p+2,r) -graph with p = 2 p=2 , namely, the distance-transitive Soicher graph with intersection array { 56 , 45 , 16 , 1 ; 1 , 8 , 45 , 56 } \{56, 45, 16, 1;1, 8, 45, 56\} , whose local graphs are isomorphic to the Gewirtz graph. The existence of an A T 4 ( p , p + 2 , r ) \mathrm {AT4}(p,p+2,r) -graph with p > 2 {p>2} remains an open question. It is known that the local graphs of each A T 4 ( p , p + 2 , r ) \mathrm {AT4}(p,p+2,r) -graph are strongly regular with parameters ( ( p + 2 ) ( p 2 + 4 p + 2 ) , p ( p + 3 ) , p − 2 , p ) \big ((p+2)(p^2+4p+2),p(p+3),p-2,p\big ) . In this paper, an upper bound is found for the prime spectrum of the automorphism group of a strongly regular graph with such parameters, and also some restrictions obtained for the prime spectrum and the structure of the automorphism group of an A T 4 ( p , p + 2 , r ) \mathrm {AT4}(p,p+2,r) -graph in the case when 2 > p 2>p is a prime power. As a corollary, it is shown that there are no arc-transitive A T ( p , p + 2 , r ) \mathrm {AT}(p,p+2,r) -graphs with p ∈ { 11 , 17 , 27 } p\in \{11,17,27\} .