Foucaud {\it et al.} recently introduced and initiated the study of a new graph-theoretic concept in the area of network monitoring. Let $G$ be a graph with vertex set $V(G)$, $M$ a subset of $V(G)$, and $e$ be an edge in $E(G)$, and let $P(M, e)$ be the set of pairs $(x,y)$ such that $d_G(x, y)\neq d_{G-e}(x, y)$ where $x\in M$ and $y\in V(G)$. $M$ is called a \emph{distance-edge-monitoring set} if every edge $e$ of $G$ is monitored by some vertex of $M$, that is, the set $P(M, e)$ is nonempty. The {\em distance-edge-monitoring number} of $G$, denoted by $\operatorname{dem}(G)$, is defined as the smallest size of distance-edge-monitoring sets of $G$. For two graphs $G,H$ of order $m,n$, respectively, in this paper we prove that $\max\{m\operatorname{dem}(H),n\operatorname{dem}(G)\} \leq\operatorname{dem}(G\,\Box \,H) \leq m\operatorname{dem}(H)+n\operatorname{dem}(G) -\operatorname{dem}(G)\operatorname{dem}(H)$, where $\Box$ is the Cartesian product operation. Moreover, we characterize the graphs attaining the upper and lower bounds and show their applications on some known networks. We also obtain the distance-edge-monitoring numbers of join, corona, cluster, and some specific networks.

翻译：Foucaud { lt et al.} 最近推出并启动了对网络监测领域新图形- 理论概念的研究。 $G$应该是一个带有顶端设置为$V( G) $( G) 的图形, $$( 美元) 的子集是$( G) 美元, $( M, e) 美元应该是一对子( x) $( y) (x, neq) d ⁇ G- e} (x, y) 美元, 其中$( 美元) m) 和$( G) 美元。 $( 美元) 是一个名为 emph{ g) 的G$ 的图形, $( 美元) 的每个边端都由某位( g) 美元( g) 的顶端点监测, 也就是, $( M) e) 的设置是非空的 。 $( 美元 ( ) 数字( ) 和 美元( 美元) 数字( 的 数字( 美元) 数字( 美元) ) 数字( 美元) 以内的 最暗( G) 表示( 美元) 數( 美元) 的數(美元) 的數( ) ) 显示的大小( ) 。