In the first part, we assign to each positive integer $n$ a digraph $\Gamma (n,5),$ whose set of vertices consists of elements of the ring ${\mathbb{Z}}_n=\{0,1,\cdots ,n-1\}$ 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 $a^5\equiv b(modn)$. Associated with $\Gamma (n,5)$ are two disjoint subdigraphs: ${\Gamma }_1(n,5)$ and ${\Gamma }_2(n,5)$ whose union is $\Gamma (n,5).$ The vertices of ${\Gamma }_1(n,5)$ are coprime to $n,$ and the vertices of ${\Gamma }_2(n,5)$ are not coprime to $n.$ In this part, we study the structure of $\Gamma (n,5)$ in detail. In the second part, we investigate the zero-divisor...

The concept of the $k$-pairable graphs was introduced by Zhibo Chen (On $k$-pairable graphs, Discrete Mathematics 287 (2004), 11–15) as an extension of hypercubes and graphs with an antipodal isomorphism. In the same paper, Chen also introduced a new graph parameter $p\left(G\right)$, called the pair length of a graph $G$, as the maximum $k$ such that $G$ is $k$-pairable and $p\left(G\right)=0$ if $G$ is not $k$-pairable for any positive integer $k$. In this paper, we answer the two open questions raised by Chen in the case that the graphs involved...

For a nonempty set $S$ of vertices in a strong digraph $D$, the strong distance $d\left(S\right)$ is the minimum size of a strong subdigraph of $D$ containing the vertices of $S$. If $S$ contains $k$ vertices, then $d\left(S\right)$ is referred to as the $k$-strong distance of $S$. For an integer $k\ge 2$ and a vertex $v$ of a strong digraph $D$, the $k$-strong eccentricity ${\mathrm{s}e}_k\left(v\right)$ of $v$ is the maximum $k$-strong distance $d\left(S\right)$ among all sets $S$ of $k$ vertices in $D$ containing $v$. The minimum $k$-strong eccentricity among the vertices of $D$ is its $k$-strong radius ${\mathrm{s}rad}_{k}D$ and the maximum...

D. Hart, A. Iosevich, D. Koh, S. Senger and I. Uriarte-Tuero (2008) showed that the distance graphs has kaleidoscopic pseudo-random property, i.e. sufficiently large subsets of d-dimensional vector spaces over finite fields contain every possible finite configurations. In this paper we study the kaleidoscopic pseudo-randomness of finite Euclidean graphs using probabilistic methods.

In this paper we derive necessary and sufficient conditions for the existence of kernels by monochromatic paths in the corona of digraphs. Using these results, we are able to prove the main result of this paper which provides necessary and sufficient conditions for the corona of digraphs to be monochromatic kernel-perfect. Moreover we calculate the total numbers of kernels by monochromatic paths, independent by monochromatic paths sets and dominating by monochromatic paths sets in this digraphs...