Symplectic Totally Isotropic Subspace Sum Graph Over Finite Fields

Graph theory and algebra are two different research fields in mathematics.In this paper,we combine graph theory and symplectic space to define the totally isotropic subspace sum graph.The connection between graph theory and symplectic space helps us apply the knowledge about symplectic space to explore the properties of graph.Let Fq be the finite field with q elements,where q is a power of prime,and S a sympletic space over Fq with dimension of 2ν.Das[7]defined the subspace sum graph of vector space,which vertex set is the set of all nontrivial subspaces in V.For any vertices W1 and W2 of g(V),if W1+W2=V,then W1 and W2 are said to be adjacent.We use g(S)to denote the totally isotropic subspace sum graph of S over Fq which vertex set is the collection of non-maximal totally isotropic subspace with dimension at least 1 of S.For any vertices W1 and W2 of g(S),if W1+W2 is a maximal totally isotropic subspace of S then W1 and W2 are said to be adjacent.Although the g(S)and the g(V)are sum graphs,they have different definitions of edges.In g(V),if two vertices W1 and W2 are adjacent then W1+W2=V,where the V is unique.In g(S),if two vertices W1 and W2 are adjacent then W1+W2 is one of the maximal totally isotropic subspaces of S.So,the results and the research methods will be different between g(V)and g(S).In chapter 1,we introduce some knowledge about symplectic space and graph theory.In chapter 2,we definite the graph g(S)and give two examples,then we prove the necessary and sufficient condition for the graph g(S)isomorphism.In chapter 3,we study some properties of the graph g(S),such as vertex degree,diameter,maximal clique,maximal independent set.In chapter 4,we obtain g(S)is eulerian and triangulated but not planar and not complete when ν≥2.