Structure of cubic mapping graphs for the ring of Gaussian integers modulo n

Yangjiang Wei; Jizhu Nan; Gaohua Tang

Czechoslovak Mathematical Journal (2012)

  • Volume: 62, Issue: 2, page 527-539
  • ISSN: 0011-4642

Abstract

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Let ℤ n [ i ] be the ring of Gaussian integers modulo n . We construct for ℤ n [ i ] a cubic mapping graph Γ ( n ) whose vertex set is all the elements of ℤ n [ i ] and for which there is a directed edge from a ∈ ℤ n [ i ] to b ∈ ℤ n [ i ] if b = a 3 . This article investigates in detail the structure of Γ ( n ) . We give suffcient and necessary conditions for the existence of cycles with length t . The number of t -cycles in Γ 1 ( n ) is obtained and we also examine when a vertex lies on a t -cycle of Γ 2 ( n ) , where Γ 1 ( n ) is induced by all the units of ℤ n [ i ] while Γ 2 ( n ) is induced by all the zero-divisors of ℤ n [ i ] . In addition, formulas on the heights of components and vertices in Γ ( n ) are presented.

How to cite

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Wei, Yangjiang, Nan, Jizhu, and Tang, Gaohua. "Structure of cubic mapping graphs for the ring of Gaussian integers modulo $n$." Czechoslovak Mathematical Journal 62.2 (2012): 527-539. <http://eudml.org/doc/246124>.

@article{Wei2012,
abstract = {Let $\mathbb \{Z\}_n\{\rm [i]\}$ be the ring of Gaussian integers modulo $n$. We construct for $\mathbb \{Z\}_n\{\rm [i]\}$ a cubic mapping graph $\Gamma (n)$ whose vertex set is all the elements of $\mathbb \{Z\}_n\{\rm [i]\}$ and for which there is a directed edge from $a \in \mathbb \{Z\}_n\{\rm [i]\}$ to $b \in \mathbb \{Z\}_n\{\rm [i]\}$ if $ b = a^3$. This article investigates in detail the structure of $\Gamma (n)$. We give suffcient and necessary conditions for the existence of cycles with length $t$. The number of $t$-cycles in $\Gamma _1(n)$ is obtained and we also examine when a vertex lies on a $t$-cycle of $\Gamma _2(n)$, where $\Gamma _1(n)$ is induced by all the units of $\mathbb \{Z\}_n\{\rm [i]\}$ while $\Gamma _2(n)$ is induced by all the zero-divisors of $\mathbb \{Z\}_n\{\rm [i]\}$. In addition, formulas on the heights of components and vertices in $\Gamma (n)$ are presented.},
author = {Wei, Yangjiang, Nan, Jizhu, Tang, Gaohua},
journal = {Czechoslovak Mathematical Journal},
keywords = {cubic mapping graph; cycle; height; cubic mapping graph; cycle; height},
language = {eng},
number = {2},
pages = {527-539},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {Structure of cubic mapping graphs for the ring of Gaussian integers modulo $n$},
url = {http://eudml.org/doc/246124},
volume = {62},
year = {2012},
}

TY - JOUR
AU - Wei, Yangjiang
AU - Nan, Jizhu
AU - Tang, Gaohua
TI - Structure of cubic mapping graphs for the ring of Gaussian integers modulo $n$
JO - Czechoslovak Mathematical Journal
PY - 2012
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 62
IS - 2
SP - 527
EP - 539
AB - Let $\mathbb {Z}_n{\rm [i]}$ be the ring of Gaussian integers modulo $n$. We construct for $\mathbb {Z}_n{\rm [i]}$ a cubic mapping graph $\Gamma (n)$ whose vertex set is all the elements of $\mathbb {Z}_n{\rm [i]}$ and for which there is a directed edge from $a \in \mathbb {Z}_n{\rm [i]}$ to $b \in \mathbb {Z}_n{\rm [i]}$ if $ b = a^3$. This article investigates in detail the structure of $\Gamma (n)$. We give suffcient and necessary conditions for the existence of cycles with length $t$. The number of $t$-cycles in $\Gamma _1(n)$ is obtained and we also examine when a vertex lies on a $t$-cycle of $\Gamma _2(n)$, where $\Gamma _1(n)$ is induced by all the units of $\mathbb {Z}_n{\rm [i]}$ while $\Gamma _2(n)$ is induced by all the zero-divisors of $\mathbb {Z}_n{\rm [i]}$. In addition, formulas on the heights of components and vertices in $\Gamma (n)$ are presented.
LA - eng
KW - cubic mapping graph; cycle; height; cubic mapping graph; cycle; height
UR - http://eudml.org/doc/246124
ER -

References

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  4. Somer, L., Křížek, M., 10.1016/j.disc.2005.12.026, Discrete Math. 306 (2006), 2174-2185. (2006) MR2255611DOI10.1016/j.disc.2005.12.026
  5. Somer, L., Křížek, M., 10.1016/j.disc.2008.04.009, Discrete Math. 309 (2009), 1999-2009. (2009) MR2510326DOI10.1016/j.disc.2008.04.009
  6. Su, H. D., Tang, G. H., The prime spectrum and zero-divisors of ℤ n [ i ] , J. Guangxi Teach. Edu. Univ. 23 (2006), 1-4. (2006) 
  7. Tang, G. H., Su, H. D., Yi, Z., Structure of the unit group of ℤ n [ i ] , J. Guangxi Norm. Univ., Nat. Sci. 28 (2010), 38-41 Chinese. (2010) 
  8. Wei, Y. J., Nan, J. Z., Tang, G. H., Su, H. D., The cubic mapping graphs of the residue classes of integers, Ars Combin. 97 (2010), 101-110 2732885. (2010) MR2732885
  9. Wei, Y. J., Nan, J. Z., Tang, G. H., 10.1007/s10587-011-0045-7, Czech. Math. J. 61 (2011), 1023-1036. (2011) MR2886254DOI10.1007/s10587-011-0045-7

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