Metric Dimension on Line Graph of Different Chemical Structures

Abstract

The Metric Dimension of a graph G is the minimum number of basis element in a resolving set. Let G = (V,E) be a connected graph and length of a shortest path between u and v is known as distance, denoted by d(u, v) in G. Let B = b1,b2,...,bk be an ordered set of vertices of G. The representation r(u|B) of u with respect to B is the k tuple (d(u,b1),d(u,b2),d(u,b3),...,d(u,bk), where B is called a resolving set or locating set if every vertex of G is uniquely identified by its distances from the vertices of B or equivalently if distinct vertices of G have distinct representations with respect to B. A resolving set of minimum cardinality is called a basis for G and cardinality is the metric dimension of G is denoted by dim(G). We investigated metric dimension on Line Graph of Polythiophene Network, Backbone Network, Hex-derive Network and Chain Sillicate Networ