ОБ АВТОМОРФИЗМАХ СИЛЬНО РЕГУЛЯРНОГО ГРАФА С ПАРАМЕТРАМИ (1276,50,0,2)

Abstract

Let \(\Gamma\) be a strongly regular graph with parameters \((v,k,0,2)\). Then \(k=u^2+1\), \(v=(u^4+3u^2+4)/2\) and \(u \equiv 1, 2, 3(mod 4)\). If \(u=1\), then \(\Gamma\) has parametrs \((4,2,0,2)\) --- tetragonal graph. If \(u=2\), then \(\Gamma\) has parametrs \((15,5,0,2)\) --- Clebsch graph. If \(u=3\), then \(\Gamma\) has parametrs \((56,10,0,2)\) --- Gewirtz graph. If \(u=5\) then hypothetical strongly regular graph\(\Gamma\) has parametrs \((352,26,0,2)\) [4]. If \(u=5\) then hypothetical strongly regular graph\(\Gamma\) has parametrs \((704,37,0,2)\) [5]. Let \(u=7\), then \(\Gamma\) has parametrs \((1276,50,0,2)\). Let \(G\) be the automorphism group of a hypothetical strongly regular graph with parameters \((1276, 50, 0, 2)\). Possible orders are found and the structure of fixed-point subgraphs is determined for elements of prime order in \(G\). With the use of theory of characters of finite groups we find the possible orders and the structures of subgraphs of the fixed points of automorphisms of the graph with parameters \((1276,50,0,2)\). It proved that if the graph with parametrs (1276,50,0,2) exist, its automorphism group divides \(2^l\cdot 3\cdot 5^m\cdot 7\cdot 11\cdot 29\). In particulary, \(G\) --- solvable group.