On Intriguing Sets In Three Classes Of Strongly Regular Graphs

Intriguing sets in strongly regular graphs which arise as collinearity graphs of finite polar spaces have been extensively studied.In this paper,we construct intriguing sets in three classes of strongly regular graphs.By the relation between eigenvalues of strongly regular graphs and parameters of intriguing sets,we know that there are two classes of intriguing sets for strongly regular graphs.The Type ?-intriguing set corresponds to positive restricted eigenvalues,and the Type ?-intriguing set corresponds to negative restricted eigenvalues.Consider the first class of strongly regular graphs.In[F.Ihringer,A.Munemasa.New strongly regular graphs from finite geometries via switching[J].Linear Algebra Appl.2019,580:464-474.],vertices are points of PG(2r,q)\Q(2r,q)which take values in nonzero squares of Fq for q=3,5.We give two constructions of Type ?-intriguing sets.The first construction uses the model V=Fq2r+1 and totally singular subspaces.The second construction uses the model V=Fqr×Fqr×Fq.We construct a subgroup K of GO(2r+1,q).The orbits obtained by the action of K on the vertex set are Type ?-intriguing sets.Consider the second class of strongly regular graphs.In[F.Ihringer,A.Munemasa.New strongly regular graphs from finite geometries via switching[J].Linear Algebra Appl.2019,580:464474.],vertices are points of PG(2r,3)\Q+(2r-1,3)which take values 1.Similar to the construction of intriguing sets for the first class of strongly regular graphs,the first construction makes use of totally singular subspaces.The second construction uses the model V=Fqr×Fqr.We construct a subgroup K of GO+(2r,q).The orbits obtained by the action of K on the vertex set are Type ?-intriguing sets.In[A.E.Brouwer,W.H.Haemers.Spectra of graphs[M].Springer.2012.],vertices of the third class of strongly regular graphs are points of PG(2r-1,q)\H(2r-1,q).Two points are adjacent if and only if the connecting line is a tangent.We construct Type?-intriguing sets for q even and r odd.By the model V=Fq2r × Fq2r,we construct a subgroup K of order qr2 of GU(2r,q).K acts on the vertex set with orbits.We can obtain Type ?-intriguing sets by the union of some orbits.