### AIMS Mathematics

2020, Issue 5: 4459-4465. doi: 10.3934/math.2020286
Research article

# On the edge metric dimension of graphs

• Received: 17 March 2020 Accepted: 10 May 2020 Published: 18 May 2020
• MSC : 05C40

• Let $G = (V, E)$ be a connected graph of order $n$. $S \subseteq V$ is an edge metric generator of $G$ if any pair of edges in $E$ can be distinguished by some element of $S$. The edge metric dimension $edim(G)$ of a graph $G$ is the least size of an edge metric generator of $G$. In this paper, we give the characterization of all connected bipartite graphs with $edim = n-2$, which partially answers an open problem of Zubrilina (2018). Furthermore, we also give a sufficient and necessary condition for $edim(G) = n-2$, where $G$ is a graph with maximum degree $n-1$. In addition, the relationship between the edge metric dimension and the clique number of a graph $G$ is investigated by construction.

Citation: Meiqin Wei, Jun Yue, Xiaoyu zhu. On the edge metric dimension of graphs[J]. AIMS Mathematics, 2020, 5(5): 4459-4465. doi: 10.3934/math.2020286

### Related Papers:

• Let $G = (V, E)$ be a connected graph of order $n$. $S \subseteq V$ is an edge metric generator of $G$ if any pair of edges in $E$ can be distinguished by some element of $S$. The edge metric dimension $edim(G)$ of a graph $G$ is the least size of an edge metric generator of $G$. In this paper, we give the characterization of all connected bipartite graphs with $edim = n-2$, which partially answers an open problem of Zubrilina (2018). Furthermore, we also give a sufficient and necessary condition for $edim(G) = n-2$, where $G$ is a graph with maximum degree $n-1$. In addition, the relationship between the edge metric dimension and the clique number of a graph $G$ is investigated by construction.

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