## Journal of Applied Mathematics

### On Simple Graphs Arising from Exponential Congruences

#### Abstract

We introduce and investigate a new class of graphs arrived from exponential congruences. For each pair of positive integers $a$ and $b$, let $G(n)$ denote the graph for which $V=\{0,1,\dots ,n-1\}$ is the set of vertices and there is an edge between $a$ and $b$ if the congruence ${a}^{x}\equiv b (\text{mod\hspace\{0.17em\}}n)$ is solvable. Let $n={p}_{1}^{{k}_{1}}{p}_{2}^{{k}_{2}}\cdots {p}_{r}^{{k}_{r}}$ be the prime power factorization of an integer $n$, where ${p}_{1}<{p}_{2}<\cdots <{p}_{r}$ are distinct primes. The number of nontrivial self-loops of the graph $G(n)$ has been determined and shown to be equal to ${\prod }_{i=1}^{r}(\varphi ({p}_{i}^{{k}_{i}})+1)$. It is shown that the graph $G(n)$ has ${2}^{r}$ components. Further, it is proved that the component ${\mathrm{\Gamma }}_{p}$ of the simple graph $G({p}^{2})$ is a tree with root at zero, and if $n$ is a Fermat's prime, then the component ${\mathrm{\Gamma }}_{\varphi (n)}$ of the simple graph $G(n)$ is complete.

#### Article information

Source
J. Appl. Math., Volume 2012 (2012), Article ID 292895, 10 pages.

Dates
First available in Project Euclid: 2 January 2013

https://projecteuclid.org/euclid.jam/1357153519

Digital Object Identifier
doi:10.1155/2012/292895

Mathematical Reviews number (MathSciNet)
MR2979465

Zentralblatt MATH identifier
1279.05074

#### Citation

Malik, M. Aslam; Mahmood, M. Khalid. On Simple Graphs Arising from Exponential Congruences. J. Appl. Math. 2012 (2012), Article ID 292895, 10 pages. doi:10.1155/2012/292895. https://projecteuclid.org/euclid.jam/1357153519