Journal of Prime Research in Mathematics

# On Metric Dimension of Chemical Networks

Muhammad Hussain$$^{a,*}$$, Saqib Nazeer$$^b$$, Hassan Raza$$^c$$

$$^a$$Department of Mathematics, COMSATS University Islamabad, Lahore Campus, Pakistan.
$$^b$$Department of Mathematics, COMSATS University Islamabad, Lahore Campus, Pakistan.
$$^c$$Department of Mathematics, COMSATS University Islamabad, Lahore Campus, Pakistan.
Correspondence should be addressed to: mmaths@gmail.com

### Abstract

Metric Dimension of any graph G is termed as the minimum number of basis element in the resolving set. Let G = (V, E) be any connected graph and length of the shortest path between s and h is known as distance, denoted by d(s, h) in G. Let B = {b1, b2, …, bq} be any ordered subset of V and representation r(u|B) with respect to B is the q−tuple (d(u, b1), d(u, b2), d(u, b3), …, d(u, bq)}, here B is called the resolving set or the locating set if every vertex of G is uniquely represented by distances from the vertices of B or if distinct vertices of G have distinct representations with respect to B. Any resolving set containing minimum cardinality is named as basis for G and its cardinality is the metric dimension of G is denoted by dim(G). We investigated metric dimension of Polythiophene Network, Backbone Network, Hex-derive Network and Nylone6,6.

#### Keywords:

Graphs, Distance, Resolving sets, Metric dimension, Chemical network.