ON AUTOMORPHISMS OF STRONGLY REGULAR GRAPH WITH THE PARAMETRS (1276,50,0,2)
https://doi.org/10.22405/2226-8383-2016-17-3-178-185
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.
About the Author
V. V. Nosov
Federal State budget institution of higher education "Orenburg State University"
Russian Federation
Russian Federation
Candidate of Physico-Mathematical Sciences, Docent, Associate Professor at the Department of Algebra and Discrete Mathematics
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Review
For citations:
Nosov V.V. ON AUTOMORPHISMS OF STRONGLY REGULAR GRAPH WITH THE PARAMETRS (1276,50,0,2). Chebyshevskii Sbornik. 2016;17(3):178-185. (In Russ.) https://doi.org/10.22405/2226-8383-2016-17-3-178-185
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