### AIMS Mathematics

2022, Issue 9: 16569-16589. doi: 10.3934/math.2022908
Research article

# The comparative study of resolving parameters for a family of ladder networks

• Received: 11 April 2022 Revised: 28 June 2022 Accepted: 04 July 2022 Published: 11 July 2022
• MSC : 05C09, 05C12, 05C92

• For a simple connected graph $G = (V, E)$, a vertex $x\in V$ distinguishes two elements (vertices or edges) $x_1\in V, y_1 \in E$ if $d(x, x_1)\neq d(x, y_1).$ A subset $Q_m\subset V$ is a mixed metric generator for $G,$ if every two distinct elements (vertices or edges) of $G$ are distinguished by some vertex of $Q_m.$ The minimum cardinality of a mixed metric generator for $G$ is called the mixed metric dimension and denoted by $dim_m(G).$ In this paper, we investigate the mixed metric dimension for different families of ladder networks. Among these families, we consider Möbius ladder, hexagonal Möbius ladder, triangular Möbius ladder network and conclude that all these families have constant-metric, edge metric and mixed metric dimension.

Citation: Mohra Zayed, Ali Ahmad, Muhammad Faisal Nadeem, Muhammad Azeem. The comparative study of resolving parameters for a family of ladder networks[J]. AIMS Mathematics, 2022, 7(9): 16569-16589. doi: 10.3934/math.2022908

### Related Papers:

• For a simple connected graph $G = (V, E)$, a vertex $x\in V$ distinguishes two elements (vertices or edges) $x_1\in V, y_1 \in E$ if $d(x, x_1)\neq d(x, y_1).$ A subset $Q_m\subset V$ is a mixed metric generator for $G,$ if every two distinct elements (vertices or edges) of $G$ are distinguished by some vertex of $Q_m.$ The minimum cardinality of a mixed metric generator for $G$ is called the mixed metric dimension and denoted by $dim_m(G).$ In this paper, we investigate the mixed metric dimension for different families of ladder networks. Among these families, we consider Möbius ladder, hexagonal Möbius ladder, triangular Möbius ladder network and conclude that all these families have constant-metric, edge metric and mixed metric dimension.

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