| Title:
|
On the tree structure of the power digraphs modulo $n$ (English) |
| Author:
|
Sawkmie, Amplify |
| Author:
|
Singh, Madan Mohan |
| Language:
|
English |
| Journal:
|
Czechoslovak Mathematical Journal |
| ISSN:
|
0011-4642 (print) |
| ISSN:
|
1572-9141 (online) |
| Volume:
|
65 |
| Issue:
|
4 |
| Year:
|
2015 |
| Pages:
|
923-945 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
For any two positive integers $n$ and $k \geq 2$, let $G(n,k)$ be a digraph whose set of vertices is $\{0,1,\ldots ,n-1\}$ and such that there is a directed edge from a vertex $a$ to a vertex $b$ if $a^k \equiv b \pmod n$. Let $n=\prod \nolimits _{i=1}^r p_{i}^{e_{i}}$ be the prime factorization of $n$. Let $P$ be the set of all primes dividing $n$ and let $P_1,P_2 \subseteq P$ be such that $P_1 \cup P_2=P$ and $P_1 \cap P_2= \emptyset $. A fundamental constituent of $G(n,k)$, denoted by $G_{P_2}^{*}(n,k)$, is a subdigraph of $G(n,k)$ induced on the set of vertices which are multiples of $\prod \nolimits _{{p_i} \in P_2}p_i$ and are relatively prime to all primes $q \in P_1$. L. Somer and M. Křížek proved that the trees attached to all cycle vertices in the same fundamental constituent of $G(n,k)$ are isomorphic. In this paper, we characterize all digraphs $G(n,k)$ such that the trees attached to all cycle vertices in different fundamental constituents of $G(n,k)$ are isomorphic. We also provide a necessary and sufficient condition on $G(n,k)$ such that the trees attached to all cycle vertices in $G(n,k)$ are isomorphic. (English) |
| Keyword:
|
congruence |
| Keyword:
|
symmetric digraph |
| Keyword:
|
fundamental constituent |
| Keyword:
|
tree |
| Keyword:
|
digraph product |
| Keyword:
|
semiregular digraph |
| MSC:
|
05C05 |
| MSC:
|
05C20 |
| MSC:
|
11A07 |
| MSC:
|
11A15 |
| MSC:
|
68R10 |
| idZBL:
|
Zbl 06537701 |
| idMR:
|
MR3441326 |
| DOI:
|
10.1007/s10587-015-0218-x |
| . |
| Date available:
|
2016-01-13T09:05:27Z |
| Last updated:
|
2020-07-03 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/144783 |
| . |
| Reference:
|
[1] Deng, G., Yuan, P.: On the symmetric digraphs from powers modulo $n$.Discrete Math. 312 (2012), 720-728. Zbl 1238.05104, MR 2872913, 10.1016/j.disc.2011.11.007 |
| Reference:
|
[2] Kramer-Miller, J.: Structural properties of power digraphs modulo $n$.Mathematical Sciences Technical Reports (MSTR) (2009), 11 pages, http://scholar.rose-hulman.edu/math\_mstr/11. |
| Reference:
|
[3] Křížek, M., Luca, F., Somer, L.: 17 Lectures on Fermat Numbers: From Number Theory to Geometry.CMS Books in Mathematics/Ouvrages de Mathématiques de la SMC 9 Springer, New York (2001). Zbl 1010.11002, MR 1866957 |
| Reference:
|
[4] Somer, L., Křížek, M.: The structure of digraphs associated with the congruence $x^k\equiv y\pmod n$.Czech. Math. J. 61 (2011), 337-358. Zbl 1249.11006, MR 2905408, 10.1007/s10587-011-0079-x |
| Reference:
|
[5] Somer, L., Křížek, M.: On symmetric digraphs of the congruence $x^k\equiv y\pmod n$.Discrete Math. 309 (2009), 1999-2009. MR 2510326, 10.1016/j.disc.2008.04.009 |
| Reference:
|
[6] Somer, L., Křížek, M.: On semiregular digraphs of the congruence $x^k\equiv y\pmod n$.Commentat. Math. Univ. Carol. 48 (2007), 41-58. MR 2338828 |
| Reference:
|
[7] Wilson, B.: Power digraphs modulo $n$.Fibonacci Q. 36 (1998), 229-239. Zbl 0936.05049, MR 1627384 |
| . |
