A NOTE ON LINE GRAPHS

Year 2017, Volume: 7 Issue: 1, 173 - 179, 01.06.2017

S. Bhavanari S. Devanaboina S. P. Kuncham

Abstract

The line graph and 1-quasitotal graph are well-known concepts in graph theory. In Satyanarayana, Srinivasulu, and Syam Prasad [13], it is proved that if a graph G consists of exactly m connected components Gi 1 ≤ i ≤ m then L G = L G1 and L G1 is the ring sum of L G2 ,..., L Gm where L G denotes the line graph of G. In [13], the authors also introduced the concept 1-quasitotal graph and obtained that Q1 G is the ring sum of G and L G where Q1 G denotes 1-quasitotal graph of a given graph G. In this note, we consider zero divisor graph of a nite associate ring R and we will prove that the line graph of Kn-1 contains the complete graph on n vertices where n is the number of elements in the ring R.

Keywords

line graph, quasi-total graph, zero-divisor graph, associate ring, complete graph.

References

• Beck, (1956), Coloring of Commutative Ring, J. Algebra, 116 (1988), 208-226.
• Bondy,J.A. and Murty,U.S.R., (1976), Graph Theory with Applications, The Macmillan Press Ltd.
• Harary,F., (1972), Graph Theory, Addison-Wesley Publishing Company, USA.
• Narsing Deo, (1997), Graph Theory with Applications to Engineering and Computer Science, Prentice Hall of India Pvt. Ltd, New Delhi.
• Armugam,S. and Ramachandran,S., (2001), Invitation to Graph Theory , Scitech Publications (India) Pvt. Ltd, Chennai.
• Kulli,V.R., (1989), Minimally Non outerplanar Graphs: A survey, (in the Book: Recent Studies in Graph theory ed: V. R. Kulli) , Vishwa International Publication, pp.177-189.
• Satyanarayana,Bh. and Syam Prasad,K., (2003), An Isomorphism Theorem on Directed Hypercubes of Dimension n, Indian J. Pure and Appl. Math, 34(10), pp.1453-1457.
• Satyanarayana,Bh. and Syam Prasad,K., (2009), Discrete Mathematics and Graph Theory, Prentice Hall India Pvt. Ltd., New Delhi, ISBN 978-81-203-3842-5.
• Satyanarayana,Bh. and Syam Prasad,K., (2014), Discrete Mathematics and Graph Theory, Prentice Hall India Pvt. Ltd., New Delhi, (Second Edition), ISBN 978-81-203-4948-3.
• Satyanarayana,Bh. and Syam Prasad,K., (2013), Nearrings, Fuzzy Ideals and Graph Theory CRC Press (Taylor and Francis Group, London, New York), ISBN 13: 9781439873106.
• Satyanarayana Bh., Syam Prasad,K., and Nagaraju,D., (2010), Prime Graph of a Ring, J. Combina- torics, Informations and System Sciences, (35), pp.27-42.
• Satyanarayana,Bh., Srinivasulu,D., and Syam Prasad,K., (2012), Some Results on Degree of Vertices in Semitotal Block Graph and Total Block Graph, International Journal of Computer Applications (0975 - 8887), Vol.50, No.9, pp.19-22.
• Satyanarayana,Bh., Srinivasulu,D., and Syam Prasad,K., (2014), Line Graphs and Quasi-Total Block Graphs, International Journal of Computer Applications (0975 - 8887) Vol.105, No.3, pp.12-16.