Let $D$ be a strongly connected oriented graph with vertex-set $V$ and arc-set $A$. The distance from a vertex $u$ to another vertex $v$, $d(u,v)$ is the minimum length of oriented paths from $u$ to $v$. Suppose $B=\{b_1,b_2,b_3,...b_k\}$ is a nonempty ordered subset of $V$. The representation of a vertex $v$ with respect to $B$, $r(v|B)$, is defined as a vector $(d(v,b_1), d(v,b_2), ..., d(v,b_k))$. If any two distinct vertices $u,v$ satisfy $r(u|B)\neq r(v|B)$, then $B$ is said to be a resolving set of $D$. If the cardinality of $B$ is minimum then $B$ is said to be a basis of $D$ and the cardinality of $B$ is called the directed metric dimension of $D$.
Let $G$ be the underlying graph of $D$ admitting a $C_n$-covering. A $C_n$-simple orientation is an orientation on $G$ such that every $C_n$ in $D$ is strongly connected. This paper deals with metric dimensions of oriented wheels, oriented fans, and amalgamation of oriented cycles, all of which admitting $C_n$-simple orientations.
Group With Maximum Undirected Edges in Directed Power Graph Among All Finite Non-Cyclic Nilpotent Groups
在[Curtin and Pourgholi，A group sum inequality and its application to power graphs，J. Algebras Combinatorics，2014]中，证明了在给定阶$ n $的群的所有有向功率图中，阶为$ n $的循环群的有向功率图具有最大数量的无向边。在本文中，我们继续他们的工作，确定了一个奇数阶的非循环幂零群$ n $，它的有向幂图在所有$n$阶的非循环幂零群中具有最多的无向边.接下来，我们确定非循环的$p$ -组，其无向功率图具有最大数量的边缘之间的所有组相同的顺序。
Key Laboratory of Intelligent Computing Signal Processing, Ministry of Education, School of Mathematical Sciences, Anhui University（安徽大学数学科学学院，教育部智能计算与信号处理重点实验室）
