The Research Of Generalized Symplectic Graph And Its Subconstituent

In this paper, we construct a graph as follows. Let (m, 0) subspaces in F_q^(2v) be the vertices, the adjacency is defined by, M - N if and only if rank(MKN^T) = 1 and dim(M∩N) = m - 1, where M - N means that M is adjacent to N, K is nonsingular alternate matrix over F_q. The graph is denoted by generalized symplectic graph relative to K over F_q. In this paper, we study the properties of generalized symplectic graph and its subconstituent. The generalized symplectic graph is a regular graph. The diameter of generalized symplectic graphis min{2m, v}, the number of vertices is (?) and the degree is(?). Forevery two vertices M and N, if rank(MKN^T) = r, dim(M∩N) = t and r + t≤m, the distance between M and N is 2m - 2t - r. In fact, the generalized symplectic graph is the generalization of symplectic graph and the dual polar graph. When m = 1, the generalized symplectic graph is the symplectic graph. When m = v. the generalized symplectic graph is the dual polar graph. In this paper, when m = 2, we also discuss the properties of the subconstituentΓ⁽¹⁾(M) of the generalized symplectic graph, where M is a special fixed vertex.Γ⁽¹⁾(M) is the union of q + 1 isomorphic connected components. When q≠2, every connected component is a strongly Deza graph with parameters (q^2(v-2)+1, (q - 1)q^2(v-2), (q - 1)²q^2(v-2)-1, (q -2)q^2(v-2). When q = 2, every connected component is a strongly Deza Graph with parameters (2^2v-3, 2^2v-4, 2^2v-5, 0).Γ⁽¹⁾(M) is a Deza Graph with parameters (3·2^2v-3, 2^2v-4, 2^2v-5, 0). At last, we give an example aboutΓ⁽¹⁾(M). When m = 2,q = 2 and v = 3,Γ⁽¹⁾(M) is a regular graph. The number of vertices ofΓ⁽¹⁾(M) is 24, the degree is 4 andΓ⁽¹⁾(M) is the union of three isomorphic connected components. Every component is not a strongly regular graph, but a strongly Deza graph with parameters (8,4, 2,0).