On subspace inclusion graph of a vector space

Abstract

In this paper, we continue the study of the subspace inclusion graph on a finite-dimensional vector space with dimension n where the vertex set is the collection of non-trivial proper subspaces of a vector space and two vertices are adjacent if one is contained in other. It is shown that is perfect and non-planar. Moreover, a necessary and sufficient condition is provided for to be Eulerian. For , it is shown that is bipartite, vertex and edge-transitive and has a perfect matching. We also provide exact value of the independence number and bounds on the domination number of for .

Keywords:

• Subspace
• symmetric graph
• q-binomial coefficient

AMS Subject Classifications:

• 05C25
• 05C69

Notes

No potential conflict of interest was reported by the author.

1 For definition, see [15].

2 It can be checked that any proper subspace is of the form of any one of the six types of subspaces.

Additional information

Funding

The research is partially funded by NBHM  Research Project Grant, (Sanction No. 2/48(10)/2013/NBHM(R.P.)/R&D II/695), Govt. of India.