Study of modified prism networks via fractional metric dimension

Abstract

For a connected network $ \Gamma $, the distance between any two vertices is the length of the shortest path between them. A vertex $ c $ in a connected network is said to resolve an edge $ e $ if the distances of $ c $ from its endpoints are unequal. The collection of all the vertices which resolve an edge is called the local resolving neighborhood set of this edge. A local resolving function is a real-valued function is defend as $ \eta : V(\Gamma) \rightarrow [0, 1] $ such that $ \eta (R_{x}(e)) \geq 1 $ for each edge $ e \in E(\Gamma) $, where $ R_{x} (e) $ represents the local resolving neighborhood set of a connected network. Thus the local fractional metric dimension is defined as $ dim_{LF}(\Gamma) = \quad min \quad \{ |\eta|: \quad \eta \quad is \quad the \quad minimal \quad local \quad resolving \quad function \quad of \quad \Gamma\}, $ where $ |\eta| = \sum \limits _ {a \in R_{x}(e)}\eta(a) $. In this manuscript, we have established sharp bounds of the local fractional metric dimension of different types of modified prism networks and it is also proved that local fractional metric dimension remains bounded when the order of these networks approaches to infinity.

Keywords

• metric dimension
• fractional metric dimension
• modified prism networks