Some Properties of the Line Graphs Associated to the Total Graph of a Commutative Ring
Pure and Applied Mathematics Journal
Volume 2, Issue 2, April 2013, Pages: 51-55
Received: Mar. 2, 2013;
Published: Apr. 2, 2013
Views 2781 Downloads 238
Aleksandra Lj. Erić, Faculty of Civil Engineering, University of Belgrade, Serbia
Zoran S. Pucanović, Faculty of Civil Engineering, University of Belgrade, Serbia
Let be a commutative ring with identity and its total graph. The subject of this article is the investigation of the properties of the corresponding line graph In particular, we determine the girth and clique number of In addition to that, we find the condition for to be Eulerian.
Aleksandra Lj. Erić,
Zoran S. Pucanović,
Some Properties of the Line Graphs Associated to the Total Graph of a Commutative Ring, Pure and Applied Mathematics Journal.
Vol. 2, No. 2,
2013, pp. 51-55.
I. Beck, "Coloring of commutative rings", J. Algebra 116, (1988) 208-226.
D.F.Anderson, P.S. Livingston, "The zero-divisor graph of a commutative ring", J. Algebra 217, (1999), 434-447.
D.F. Anderson, A. Badawi, "The total graph of a commutative ring", J. Algebra 2008, 320, 2706-2719.
H.R. Maimani, C. Wickham, S. Yassemi, "Rings whose total graph have genus at most one", Rocky Mountain J. Math., to appear.
H. Whitney, "Congruent graphs and connectivity of graphs", American Journal of Math. 54 (1932), 150-168.
Hung-Jen Chiang-Hsieh, Pei-Feng Lee, Hsin-Ju Wang, "The embedding of line graphs associated to the zero-divisor graphs of commutative rings", Israel J. Math. 180, (2010) 193-222.
Z. Petrović, Z. Pucanović, "The line graph associated to the total graph of a commutative ring", Ars Combinatoria, to appear.
I. Kaplansky, " Commutative rings", Revised Edition, University of Chicago Press, Chicago, 1974.
M.F. Atiyah, I.G. Macdonald, "Introduction to commutative algebra", Adison-Wesley Publishing Company, Reading, MA. 1969.
D.B. West, "Introduction to graph theory", Prantice Hall, Inc., Upper Saddle River, NJ, 1996.
R. Diestel, "Graph Theory", Third Edition, Graduate Texts in Mathematics 173, Springer-Verlag Berlin Heidelberg, 2005