On Metric Dimension in Some Hex Derived Networks

Area of Science:

• Graph Theory
• Network Science
• Robotics

Background:

• The metric dimension is a graph invariant used to model systems where location identification is crucial.
• Robot navigation requires efficient methods for agents to determine their position within a network.
• Existing models may not fully capture the dynamic nature of agent movement in complex networks.

Purpose of the Study:

• To determine the metric dimension of hex derived networks (HDN1(n)).
• To establish the minimum number of nodes required for unique identification of all nodes in HDN1(n).
• To provide insights into the network's suitability for applications like robot navigation.

Main Methods:

• Utilized graph theory principles to define and analyze the metric dimension.
• Applied specific mathematical formulations to the structure of HDN1(n) graphs.
• Proved that the metric dimension for HDN1(n) with n ≥ 2 is exactly four.

Main Results:

• The metric dimension of HDN1(n) for n ≥ 2 was calculated as m d ( H D N 1 ( n ) ) = 4.
• Demonstrated that four nodes are sufficient to uniquely identify every node within these networks.
• This result provides a concrete value for network resolvability.

Conclusions:

• The metric dimension of hex derived networks (HDN1(n)) is four for n ≥ 2.
• Four nodes are the minimum required for a complete set of landmark nodes in these networks.
• This has implications for the design and efficiency of sensor networks and robot navigation systems.