On a Connection of Number Theory with Graph Theory

Somer, Lawrence; Křížek, Michal

doi:10.1023/b:cmaj.0000042385.93571.58

Cited by 36 publications

(29 citation statements)

References 9 publications

(10 reference statements)

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“…The paper extends results given in the works [7], [10], [14], and [16], which provide an interesting connection between number theory, graph theory, and group theory. In the papers [10]- [13], we investigated properties of the iteration digraph representing a dynamical system occurring in number theory.…”

Section: Introductionsupporting

confidence: 84%

“…It was proved in [10] that if k = 2 then G(n, k) has a nonzero isolated fixed point if and only if n = 2m, where m is an odd square-free integer. In this case, a is a nonzero isolated fixed point if and only if a = m. In Theorem 8.2, we extend this result by counting isolated fixed points in G(n, k) for any n > 1 and any k 2.…”

Section: Results On Fixed Pointsmentioning

confidence: 99%

“…In the papers [10]- [13], we investigated properties of the iteration digraph representing a dynamical system occurring in number theory. For related results also see [1].…”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

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

Somer

Křížek

2011

Czech Math J

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The structure of digraphs associated with the congruence x k ≡ y (mod n) Czechoslovak Mathematical Journal, Vol. 61 (2011) Abstract. 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.

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Section: Introductionsupporting

confidence: 84%

Section: Results On Fixed Pointsmentioning

confidence: 99%

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

Somer

Křížek

2011

Czech Math J

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“…We recall that a directed graph is a finite set of vertices together with directed edges. The iteration digraph of a map f : S → S on a finite set S is a directed graph, whose vertices are elements of S and whose directed edges connect each x ∈ S with its image f (x) ∈ S. The iteration graphs of the function f (x) = x k on the rings S = Z n have interesting connections to number theory and have been extensively discussed (see [8]- [12]). These digraphs reflect the properties of Z n and f. For each positive integer n, we denote such an iteration graph on the ring Z n by Γ(n, k).…”

Section: Introductionmentioning

confidence: 99%

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

Teng-xia

2014

Czech Math J

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“…If R is any set, a mapping f : R → R induces a directed graph on R, called the iteration digraph of f , whose vertices are the elements of R and whose directed edges connect each x ∈ R with its image f (x) ∈ R. The iteration graphs of the squaring function f (x) = x 2 on the rings R = /n have interesting connections to number theory (see, e.g., [6] and [2]) and have been extensively studied (see, e.g., [5], [8], [1], and [6]), yet interesting questions about them remain unanswered. For each positive integer n, we denote the iteration graph of the squaring function on the ring /n by G(n).…”

Section: Introductionmentioning

confidence: 99%