Non-zero component union graph of a finite-dimensional vector space

Abstract

In this paper, we introduce a graph structure, called non-zero component union graph on finite-dimensional vector spaces. We show that the graph is connected and find its domination number, clique number and chromatic number. It is shown that two non-zero component union graphs are isomorphic if and only if the base vector spaces are isomorphic. In case of finite fields, we study the edge-connectivity and condition under which the graph is Eulerian. Moreover, we provide a lower bound for the independence number of the graph. Finally, we come up with a structural characterization of non-zero component union graph.

Keywords:

• Basis
• independent set
• graph

AMS Subject Classifications:

• 05C25
• 05C69

Acknowledgements

The author is thankful to the anonymous reviewer for suggestions regarding the characterization given in Section 8.

Notes

No potential conflict of interest was reported by the author.

1 Clearly, (Note that if , then I cannot be an independent set.)

2 For definition, see Section 2.

Additional information

Funding

The work was supported by NBHM Research Project Grant, [Sanction No. 2/48(10)/2013/NBHM(R.P.)/R&D II/695], Govt. of India.