Strongly Regular Graphs With 4-vertex Condition

In this thesis,we use the construction method to construct strongly regular graphs ?c by groups G,and some properties of the strongly regular graphs ?G are obtained,the necessary and sufficient condition for the scheme XG is schurian proved by combining with the theory of coherent configuration.For any group G(IGI=m),we can construct strongly regular graph ?G.On this basis,we further study when G is the direct product of group H and K,the strongly regular graphs satisfies(?).Then two conclusions are obtained:·Aut?G is a primitive permutation groups of rank 3 if and only if G is an elementary abelian 2-group with order greater than or equal to 4 or a cyclic group of order 5.·For a geometric graph ? with characteristic(3,m,2),if its automorphism group is a primitive permutation groups of rank 3,we can get ? is isomorphic to ?G,where G is an elementary abelian 2-group with order greater than or equal to 4 or a cyclic group of order 5.Finally,we obtain a sufficient and necessary condition that strongly regular graph?G satisfies the 4-vertex condition is equivalent to the scheme XG is schurian:·For group G(|G|?4),let XG be a scheme from strongly regular graphs ?G,then XG is schurian iff ?G satisfies the 4-vertex condition.