Displaying similar documents to “On iteration digraph and zero-divisor graph of the ring ℤ n ”

Properties of digraphs connected with some congruence relations

J. Skowronek-Kaziów (2009)

Czechoslovak Mathematical Journal

Similarity:

The paper extends the results given by M. Křížek and L. Somer, , Czech. Math. J. 54 (129) (2004), 465–485 (see [5]). For each positive integer n define a digraph Γ ( n ) whose set of vertices is the set H = { 0 , 1 , ⋯ , n - 1 } and for which there is a directed edge from a ∈ H to b ∈ H if a 3 ≡ b ( mod n ) . The properties of such digraphs are considered. The necessary and the sufficient condition for the symmetry of a digraph Γ ( n ) is proved. The formula for the number of fixed points of Γ ( n ) is established. Moreover, some connection of the length...

The cubic mapping graph for the ring of Gaussian integers modulo n

Yangjiang Wei, Jizhu Nan, Gaohua Tang (2011)

Czechoslovak Mathematical Journal

Similarity:

The article studies the cubic mapping graph Γ ( n ) of ℤ n [ i ] , the ring of Gaussian integers modulo n . For each positive integer n > 1 , the number of fixed points and the in-degree of the elements 1 ¯ and 0 ¯ in Γ ( n ) are found. Moreover, complete characterizations in terms of n are given in which Γ 2 ( n ) is semiregular, where Γ 2 ( n ) is induced by all the zero-divisors of ℤ n [ i ] .

On a connection of number theory with graph theory

Lawrence Somer, Michal Křížek (2004)

Czechoslovak Mathematical Journal

Similarity:

We assign to each positive integer n a digraph whose set of vertices is H = { 0 , 1 , ⋯ , n - 1 } and for which there is a directed edge from a ∈ H to b ∈ H if a 2 ≡ b ( m o d n ) . We establish necessary and sufficient conditions for the existence of isolated fixed points. We also examine when the digraph is semiregular. Moreover, we present simple conditions for the number of components and length of cycles. Two new necessary and sufficient conditions for the compositeness of Fermat numbers are also introduced.

Balanced path decomposition of λ K n , n and λ K n , n *

Hung-Chih Lee, Chiang Lin (2009)

Czechoslovak Mathematical Journal

Similarity:

Let P k denote a path with k edges and ł K n , n denote the ł -fold complete bipartite graph with both parts of size n . In this paper, we obtain the necessary and sufficient conditions for ł K n , n to have a balanced P k -decomposition. We also obtain the directed version of this result.

On semiregular digraphs of the congruence x k ≡ y ( mod n )

Lawrence Somer, Michal Křížek (2007)

Commentationes Mathematicae Universitatis Carolinae

Similarity:

We assign to each pair of positive integers n and k ≥ 2 a digraph G ( n , k ) whose set of vertices is H = { 0 , 1 , ⋯ , n - 1 } and for which there is a directed edge from a ∈ H to b ∈ H if a k ≡ b ( mod n ) . The digraph G ( n , k ) is semiregular if there exists a positive integer d such that each vertex of the digraph has indegree d or 0. Generalizing earlier results of the authors for the case in which k = 2 , we characterize all semiregular digraphs G ( n , k ) when k ≥ 2 is arbitrary.