Displaying similar documents to “On a connection of number theory with graph theory”

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...

On iteration digraph and zero-divisor graph of the ring ℤ n

Tengxia Ju, Meiyun Wu (2014)

Czechoslovak Mathematical Journal

Similarity:

In the first part, we assign to each positive integer n a digraph Γ ( n , 5 ) , whose set of vertices consists of elements of the ring ℤ n = { 0 , 1 , ⋯ , n - 1 } with the addition and the multiplication operations modulo n , and for which there is a directed edge from a to b if and only if a 5 ≡ b ( mod n ) . Associated with Γ ( n , 5 ) are two disjoint subdigraphs: Γ 1 ( n , 5 ) and Γ 2 ( n , 5 ) whose union is Γ ( n , 5 ) . The vertices of Γ 1 ( n , 5 ) are coprime to n , and the vertices of Γ 2 ( n , 5 ) are not coprime to n . In this part, we study the structure of Γ ( n , 5 ) in detail. In the second part, we investigate...

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 ] .

The structure of digraphs associated with the congruence x k ≡ y ( mod n )

Lawrence Somer, Michal Křížek (2011)

Czechoslovak Mathematical Journal

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 ) . We investigate the structure of G ( n , k ) . In particular, upper bounds are given for the longest cycle in G ( n , k ) . We find subdigraphs of G ( n , k ) , called fundamental constituents of G ( n , k ) , for which all trees attached to cycle vertices are isomorphic.