A Subspace Inclusion Graph of a Finite Dimensional Vector Space

The combination of algebraic structures and graphs was carried out by investigating thoroughly the relation among the algebraic structure and the graph theoretic properties. Moreover, it needs to explore algebraic structure. The results of the combination of algebraic structures and graphs have many applications in the fields of Internet modeling, coding, etc. For example, the famous Cayley graph was constructed from groups and widely used in network models. Das introduced the subspace inclusion graph on finite-dimensional vector space over a finite filed, and studied that the graph is bipartite and some special properties if the dimension of the vector space is 3. In this paper, this bipartite inclusion graph in the case of 3-dimensional is extended to more general dimensional bipartite inclusion graph. The diameter, girth, clique number, covering number, independence number and matching number are studied and the properties are shown, such as regular, planar and Eulerian. Moreover, the authors also introduce a new results about the structure and the number of 1-dimensional and n-1-dimensional subspaces on n-dimensional vector space.