Computing Bounds of Fractional Metric Dimension of Metal Organic Graphs

Metal organic graphs are hollow structures of metal atoms that are connected by ligands, where metal atoms are represented by the vertices and ligands are referred as edges. A vertex x resolves the vertices u and v of a graph G if d u , x ≠ d v , x . For a pair u , v of vertices of G , R u , v = x ∈ V G : d x , u ≠ d x , v is called its resolving neighbourhood set. For each pair of vertices u and v in V G , if f R u , v ≥ 1 , then f from V G to the interval 0,1 is called resolving function. Moreover, for two functions f and g , f is called minimal if f ≤ g and f v ≠ g v for at least one v ∈ V G . The fractional metric dimension (FMD) of G is denoted by dim f G and defined as dim f G = min g : g   is a minimal resolving function of  G , where g = ∑ v ∈ V G g v . If we take a pair of vertices u , v of G as an edge e = u v of G , then it becomes local fractional metric dimension (LFMD) dim l f G . In this paper, local fractional and fractional metric dimensions of MOG n are computed for n ≅ 1 mod 2 in the terms of upper bounds. Moreover, it is obtained that metal organic is one of the graphs that has the same local and fractional metric dimension.