Constructing Symplectic Graphs With 2 Dimensional Totally Isotropic And Non-isotropic Subspaces In Symplectic Space Over Finite Fields

The theory of geometry of classical groups over finite fields is a very important class of algebraic and geometric structures.There are a lot of important applications,such as association schemes,geometry lattices,algebraic codes and graphs theory.In this thesis,we mainly investigate its application in constructing symplectic graph.The main results are as follows:1.As a generalization of dipole graphs,we construct a symplectic graph Γ whose vertex set is all 2 dimensional totally isotropic subspaces of the symplectic space,and two vertices are adjacent if and only if their intersection is a 1 dimensional subspace.We calculate the degree of vertex and the number of common neighbors of two distinct vertices of Γ,and prove that Γ is a d–Deza graph.In addition,we study the regularity of the subconstituents of Γ and calculate the number of common neighbors of two distinct vertices of the subconstituents.Furthermore,we define the spanning subgraph H of symplectic graph Γ by increasing the adjacency condition,such that the number of common neighbors of two adjacent vertices of H is fixed.That is to say,H is a quasi–strongly regular graph.2.We construct a symplectic graph G whose vertex set is all 2 dimensional non-isotropic subspaces of the symplectic space,and two vertices are adjacent if and only if their intersection is a 1 dimensional subspace.We prove that G is a quasi–strongly regular graph,by calculating the degree of vertex and the number of common neighbors of two distinct vertices of G.And second,we study the structure of the subconstituents of G.