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In this paper an enumeration method to find the number of triangles in the zero-divisor Cayley graph G(Zₙ,D₀ ) associated with the ring (Zₙ,⨁,⨀),n≥1 of integers modulo n,  an integer and its subset D0 of zero-divisors is presented. Further it is shown that this graph is Hamiltonian, not bipartite  and  Eulerian graph when n is odd.

 

References

  1. Nathanson MB. Components of Arithmetic graphs. Monatsheftefu ̈r Mathematik. 1980; 29: 219-220.
     Google Scholar
  2. Dejter IJ, Giudici RE. On unitary Cayley graphs. JCMCC. 1995; 18: 121-124.
     Google Scholar
  3. Bierrizbeitia P, Giudici RE. Counting pure k-cycles in sequences of Cayley graphs. Discrete Math. 1996; 149: 11-18.
     Google Scholar
  4. Bierrizbeitia P, Giudici RE. On cycles in the sequence of unitary Cayley graphs. Reports Techico No. 01-95, Universidad Simon Bolivear, Depto. De Mathematics, Caracas, Venezuela. 1995.
     Google Scholar
  5. Madhavi L. Studies on domination parameters and Enumeration of cycles in some arithmetic graphs, Ph.D Thesis, Sri Venkateswara University, India. 2003.
     Google Scholar
  6. Chalapati T, Madhavi L. Enumeration of Triangles in a Divisor Cayley Graph. Momona Ethiopian Journal of Science, 2013; V(5) :163-173.
     Google Scholar
  7. Maheswari B, Madhavi L. Enumeration of Hamilton Cycles and Triangles in Euler totient Cayley Graphs, Graph Theory Notes of New York LIX. 2010; 28-31.
     Google Scholar
  8. Maheswari B. Madhavi L. Enumeration of Triangles and Hamilton Cycles in Quadratic Residue Cayley Graphs. Chamchuri Journal of Mathematics. 2009; 1: 95-103.
     Google Scholar
  9. Beck I. Coloring of commutative rings. J. Algebra. 1998; 116: 208-206.
     Google Scholar
  10. Anderson D, Naseer M. Beck’s Coloring of Commutative Ring. J. Algebra. 1993; 159: 500-514.
     Google Scholar
  11. Anderson DF, Livingston PS. The zero-divisor graph of commutative ring. J. Algebra. 1999; 217: 434-447.
     Google Scholar
  12. Smith NO. Planar Zero-Divisor Graph. International Journal of Commutative Rings. 2002; 2(4): 177-188.
     Google Scholar
  13. Tongsuo W. On Directed Zero-Divisor Graphs of Finite Rings. Discrete Mathematics. 2005; 296(1): 73-86.
     Google Scholar
  14. Devendra J. Nagalakshumma T, Madhavi L. The radius, diameter, girth and circumference the of zero-divisor Cayley graph of the residue class ring (Z_n,⨁,⨀). IOSR Journal of Mathematics. 2019; 15(4): 58-63.
     Google Scholar
  15. Devendra J. Madhavi L, Nagalakshumma T. The zero-divisor Cayley graph of the residue class ring (Z_n,⨁,⨀). Malaya Journal of Mathematik. 2019; 7(3): 590-594.
     Google Scholar
  16. Madhavi L, Devendra J, Nagalakshumma T. The zero-divisor Cayley graph of the residue class ring (Z_n,⨁,⨀). Journal of Computer and Mathematical Sciences. 2019; 10(9): 1589-1567.
     Google Scholar
  17. Bondy JA, Murty USR. Graph Theory with Applications. Macmillan, London. 1979: 290.
     Google Scholar
  18. Gallian JA. Contemporary Abstract Algebra. Narosa publishing house, Ninth edition. 2018.
     Google Scholar
  19. Apostol TM. Introduction to Analytic Number Theory. Springer International Student Edition. 1989.
     Google Scholar
  20. Livingston PS. structure in zero-divisor Graphs of commutative rings. M. S. Thesis, The University of Tennessee, Knoxville, TN; 1997.
     Google Scholar