Totally Isotropic Subspace Inclusion Graph Of A Unitary Space

Graph theory and algebra have always been two important research fields in mathematics.Although they have their own different research contents,they are inextricably linked.On the one hand,the properties of graph theory can be used to characterize some algebraic structures.On the other hand,algebraic theory is often used to solve some problems of graph theory.Let Fq2 be a finite field with q2 elements,where q is a power of a prime.We combine unitary space and graph to construct the totally isotropic subspace inclusion graph of the unitary space U over Fq2 denoted by In(U).The vertex set is the collection of all totally isotropic subspaces with dimension at least 1 of U.For any two vertices U1 and U2 of In(U),if U1(?)U2 or U2(?)U1 holds,then U1 and U2 are said to be adjacent,that is,U1?U2.Obviously,In(U)is a subgraph of In(V).Note that these two graphs have different vertex sets.We know any two vertices of In(V)have at least one common neighbour.However,there exists two different vertices which have no common neighbours in In(U).So the results and the research methods in this paper will be different from the former.In chapter 1,we introduce some preliminary knowledge.In chapter 2,we study some properties of the graph In(U),such as diameter,girth,clique number and chromatic number.Furthermore,we give the necessary and sufficient conditions for In(U)to be triangulated.Then we prove that two unitary spaces are isomorphic if and only if their corresponding totally isotropic subspace inclusion graphs are isomorphic.In chapter 3,we obtain In(U)is not complete,and In(U)is not planar when the dimension of the unitary space U is greater than or equal to 6.In addition,we give the necessary and sufficient conditions for In(U)to be a bipartite graph,regular graph and Euler graph,respectively.