Fractional Metric Dimension of Different Chemical Structures

Abstract

The Metric Dimension of a graph G is the minimum number of basis element in a resolving set. Let Q= {q1, q2, q3,…, qk} is the set of vertices of G. The representation r(u/Q) of u with respect to Q is the k-tuple {d(u,q1 ), d(u,q2 ), d(u,q3 ), …, d(u,qk )}, where Q is called a resolving set [4], if every vertex of G is uniquely identified by its distances from the vertices of Q, or equivalently, if distinct vertices of G have distinct representations with respect to Q. Minimum number of vertices in a resolving set is called the basis for G and the number of basis elements is known as the metric dimension of G denoted by dim (G) [10]. In recent years some mathematics work on the concept of resolving sets and metric basis [1, 7, 13]. We have an ordered set of vertices Q ={q1, q2,…, qk } of a graph G, the d(u,qi ) is zero iff u= qi. If r(s/Q) ≠ r(t/Q) for each pair of distinct vertices s,t belongs to V(G)\Q then Q is called a resolving set. The Fractional Metric Dimension of G is defined as dim f(G) = {|g|: g is minimal Resolving function of G} Where |g| = ∑ V∈Vg (v). In the graph theory the most important research study in integer-valued graph theoretic concept of fractionalization. The research is about different chemical structure and convert these structure into planar network graphs. Consequently to calculate the fractional metric dimension of these network graphs. In this thesis, we investigated metric dimension and fractional metric dimension of Line Graph of Polythiophene Network and Backbone Network.