On the tree structure of the power digraphs modulo $n$

Sawkmie, Amplify, and Singh, Madan Mohan. "On the tree structure of the power digraphs modulo $n$." Czechoslovak Mathematical Journal 65.4 (2015): 923-945. <http://eudml.org/doc/276196>.

@article{Sawkmie2015,
abstract = {For any two positive integers $n$ and $k \ge 2$, let $G(n,k)$ be a digraph whose set of vertices is $\lbrace 0,1,\ldots ,n-1\rbrace $ and such that there is a directed edge from a vertex $a$ to a vertex $b$ if $a^k \equiv b \hspace\{4.44443pt\}(\@mod \; n)$. 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.},
author = {Sawkmie, Amplify, Singh, Madan Mohan},
journal = {Czechoslovak Mathematical Journal},
keywords = {congruence; symmetric digraph; fundamental constituent; tree; digraph product; semiregular digraph},
language = {eng},
number = {4},
pages = {923-945},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {On the tree structure of the power digraphs modulo $n$},
url = {http://eudml.org/doc/276196},
volume = {65},
year = {2015},
}

TY - JOUR
AU - Sawkmie, Amplify
AU - Singh, Madan Mohan
TI - On the tree structure of the power digraphs modulo $n$
JO - Czechoslovak Mathematical Journal
PY - 2015
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 65
IS - 4
SP - 923
EP - 945
AB - For any two positive integers $n$ and $k \ge 2$, let $G(n,k)$ be a digraph whose set of vertices is $\lbrace 0,1,\ldots ,n-1\rbrace $ and such that there is a directed edge from a vertex $a$ to a vertex $b$ if $a^k \equiv b \hspace{4.44443pt}(\@mod \; n)$. 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.
LA - eng
KW - congruence; symmetric digraph; fundamental constituent; tree; digraph product; semiregular digraph
UR - http://eudml.org/doc/276196
ER -