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Domination number of total graphs

  • Abbas Shariatinia , Hamid Reza Maimani and Siamak Yassemi EMAIL logo
Published/Copyright: December 30, 2016
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Mathematica Slovaca
From the journal Mathematica Slovaca Volume 66 Issue 6

Abstract

Let R be a commutative ring with Z(R) the set of zero-divisors and U(R) the set of unit elements of R. The total graph of R, denoted by T(Γ(R)), is the (undirected) graph with all elements of R as vertices, and for distinct x, y ∈ R, the vertices x and y are adjacent if and only if x + y ∈ Z(R). We study the domination number of T(Γ(R)). It is shown that if R = Z(R) ∪ U(R), then the domination number of T(∪(R)) is finite provided R has a maximal ideal of finite index. Moreover, if R=∏i=1nFi, where Fi is a field for each 1 ≤ i ≤ n and t = |F1| ≤ |F2| ≤ ··· ≤ |Fn|, then the domination number of T(Γ(R)) is equal to t - 1 provided t = |Fi| for every 1 ≤ i ≤ n, and is equal to t otherwise. Finally, for an R-module M it is shown that the total domination number of the total graph of the idealization (Nagata extension) R(+)M is equal to the domination number of the total graph of R provided M is a torsion free R-module or R = Z(R) ∪ U(R).

MSC 2010: Primary 05C75; Secondary 13M05
Keywords: total graph; domination number; idealization (Nagata extension)

The research of H. R. Maimani was in part supported by a grant from IPM (No. 93050112).

(Communicated by Miroslav Ploščica)

Acknowledgement

The authors are grateful to the anonymous referee for making many constructive suggestions. Part of this work was done while S. Yassemi was visited the Max Planck Institute for Mathematics (MPIM), Bonn, Germany. He would like to thank MPIM for sponsoring his visit to Bonn in 2014.

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Received: 2013-12-23
Accepted: 2015-1-12
Published Online: 2016-12-30
Published in Print: 2016-12-1

© 2016 Mathematical Institute Slovak Academy of Sciences

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https://doi.org/10.1515/ms-2016-0241
Keywords for this article
total graph; domination number; idealization (Nagata extension)

Articles in the same Issue

  2. A triple representation of lattice effect algebras
  4. Periods of morgan-voyce sequences and elliptic curves
  6. Multiplicative generalized derivations on ideals in semiprime rings
  8. A few remarks on Poincaré-Perron solutions and regularly varying solutions
  10. Applications of extremal theorem and radius equation for a class of analytic functions
  12. On the “bang-bang” principle for a class of Riemann-Liouville fractional semilinear evolution inclusions
  14. New criteria for global exponential stability of linear time-varying volterra difference equations
  16. On approximation of functions by some hump matrix means of Fourier series
  18. Pseudo-amenability and pseudo-contractibility for certain products of Banach algebras
  20. Poisson kernels on semi-direct products of abelian groups
  22. Locally convex projective limit cones
  24. On the properties (wL) and (wV)
  26. On the existence of solutions for quadratic integral equations in Orlicz spaces
  28. Existence and uniqueness of best proximity points under rational contractivity conditions
  30. Almost Weyl structures on null geometry in indefinite Kenmotsu manifolds
  32. On the internal approach to differential equations 3. Infinitesimal symmetries
  34. A Characterization of the discontinuity point set of strongly separately continuous functions on products
  36. Wick differential and Poisson equations associated to the 𝚀𝚆𝙽-Euler operator acting on generalized operators
  38. Multivariate EIV models
  40. On codes over 𝓡k, m and constructions for new binary self-dual codes
  42. Domination number of total graphs
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