Document Type : Original Manuscript

Authors

1 Department of Pure Mathematics, Faculty of Mathematical Sciences, University of Guilan, P.O. Box 19141, Rasht, Iran.

2 Department of Pure Mathematics, Faculty of Mathematical Sciences, University of Guilan, P.O. Box 19141, Rasht, Iran.Guilan, Rasht, Iran.

3 Center of Excellence for Mathematical Modeling, Optimization and Combinatorial Computing (MMOCC), University of Guilan, Rasht, Iran.

Abstract

We give a decomposition of total graphs on some finite commutative rings R = Zm, where the set of zero-divisors of R is not an ideal. In particular, we study the total graph T(􀀀(Z2npm))
where p is a prime and m and n are positive integers and investigate some graph theoretical properties with some of its fundamental subgraphs.

Keywords

[1] R. B. Allan and R. Laskar, On domination and independent domination numbers of a graph, Discrete Math., 23 (1978), 73–76.
[2] D. F. Anderson and A. Badawi, The total graph of a commutative ring, J. Algebra, 320 (2008), 2706–2719.
[3] M. Behzad, G. Chartrand and J. K. Cooper, The colour numbers of complete graphs, J. London Math. Soc., 42 (1967), 226–228.
[4] M. H. Shekarriz, M. H. Shirdareh Haghighi and H. Sharif, On the Total graph of a finite commutative ring, Comm. Algebra, 40(8) (2012), 2798–2807.
[5] D. B. West, Introduction to Graph Theory, Second edition, Prentice-Hall, Upper Saddle river, NJ, 1996.