The fractional metric dimension of graphs

A vertex xx in a connected graph GG is said to resolve a pair {u,v}{u,v} of vertices of GG if the distance from uu to xx is not equal to the distance from vv to xx. A set SS of vertices of GG is a resolving set for GG if every pair of vertices is resolved by some vertex of SS. The smallest cardinality of a resolving set for GG, denoted by dim(G)dim(G), is called the metric dimension of GG. For the pair {u,v}{u,v} of vertices of GG the collection of all vertices which resolve the pair {u,v}{u,v} is denoted by R{u,v}R{u,v} and is called the resolving neighbourhood of the pair {u,v}{u,v}. A real valued function g:V(G)→[0,1]g:V(G)→[0,1] is a resolving function of GG if g(R{u,v})≥1g(R{u,v})≥1 for any two distinct vertices u,v∈V(G)u,v∈V(G). The fractional metric dimension of GG is defined as dimf(G)=min{|g|:g is a minimal resolving function of G}dimf(G)=min{|g|:g is a minimal resolving function of G}, where |g|=∑v∈Vg(v)|g|=∑v∈Vg(v). In this paper we study this parameter.

► First paper presenting basic results on fractional metric dimension. ► Upper and lower fractional metric independence number—new concepts introduced in this paper. ► Fractional metric dimension of standard graphs. ► Fundamental results on graphs with fractional metric dimension n2 where nn is the order of the graph.