Abstract
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 is an element of 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) not equal g (v) for at least one v is an element of V (G). The fractional metric dimension (FMD) of G is denoted by dim(f)(G) and defined as dim(f)(G) = min{vertical bar g vertical bar: g is a minimal resolving function of G}, where vertical bar g&VERBAR = Sigma(v is an element of V(G))g(v). If we take a pair of vertices (u, v) of G as an edge e = uv of G, then it becomes local fractional metric dimension (LFMD) (dim(lf) (G)). In this paper, local fractional and fractional metric dimensions of MOG (n) are computed for n (sic) 1 (mod2) 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.