The Simultaneous Fractional Dimension of Graph Families

For a connected graph G with vertex set V , let R G x, y = z ∈ V : d G ( x, z ) ≠ d G ( y, z ) for any distinct x, y ∈ V , where d G ( u, w ) denotes the length of a shortest uw -path in G . For a real-valued function g defined on V , let g ( V ) = ∑ s ∈ V g ( s ). Let C = {G_1,G_2, … ,G_k} be a family of connected graphs having a common vertex set V , where k ≥ 2 and ∣ V ∣≥ 3. A real-valued function h : V → [0, 1] is a simultaneous resolving function of C if h ( R G x, y ) ≥ 1 for any distinct vertices x, y ∈ V and for every graph G ∈ C . The simultaneous fractional dimension , Sd_f( C) , of C is min h ( V ): h is a simultaneous resolving function of C . In this paper, we initiate the study of the simultaneous fractional dimension of a graph family. We obtain max _1 ≤ i ≤ k{ _f(G_i)}≤Sd_f( C) ≤min{∑_i = 1^k _f(G_i),|V| 2 , where both bounds are sharp. We characterize C satisfying Sd_f( C) = 1 , examine C satisfying Sd_f( C) = |V| 2 , and determine Sd_f( C) when C is a family of vertex-transitive graphs. We also obtain some results on the simultaneous fractional dimension of a graph and its complement.