African Journal of
Mathematics and Computer Science Research

  • Abbreviation: Afr. J. Math. Comput. Sci. Res.
  • Language: English
  • ISSN: 2006-9731
  • DOI: 10.5897/AJMCSR
  • Start Year: 2008
  • Published Articles: 240

Full Length Research Paper

Some results on principal ideal graph of a ring

Satyanarayana Bhavanari1, Godloza Lungisile2 and Nagaraju Dasari3*
  1Department of Mathematics, Acharya Nagarjuna University, Nagarjuna Nagar – 522 510, AP, India. 2Department of Mathematics, Walter Sisulu University, Umtata, South Africa. 3Department of Science and Humanities (Maths), HITS, Hindustan University, Padur, Chennai – 603 103, India.
Email: [email protected]

  • Article Number - 727C6A02532
  • Vol.3(6), pp. 235 - 241, June 2011
  •  Accepted: 31 May 2011
  •  Published: 30 June 2011

Abstract

 

Let R be an associative (not necessarily commutative) ring. In the present paper, we introduced a new type of graph (called ‘Principal Ideal Graph’, denoted by PIG(R)) related to a given associative ring R.  We presented some examples. We obtained few fundamental important relations between rings and graphs with respect to the properties: simple ring, complete graph, etc. We also observed that if R and S are isomorphic rings, then the related principal ideal graphs are isomorphic, but the converse is not true. We defined an equivalence relation on a given ring R and obtained a one-to-one correspondence between the set of all equivalence classes and the set of all connected components of PIG(R). We introduced the concept ‘full Hamiltonian decomposition’ for a general graph, and proved that there exists a full Hamiltonian decomposition for PIG(R).

 

Key words: Principal ideal graph, ring, Hamiltonian decomposition, complete graph.

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