The Simultaneous Fractional Dimension of Graph Families

Abstract

For a connected graph G with vertex set V, let RG{x, y} = {z ∈ V: dG(x, z) ≠ dG(y, z)} for any distinct x, y ∈ V, where dG(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∈Vg(s). Let $${\cal C} = \{{G_1},{G_2}, \ldots ,{G_k}\} $$ 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 $${\cal C}$$ if h(RG{x, y}) ≥ 1 for any distinct vertices x, y ∈ V and for every graph $$G \in {\cal C}$$ . The simultaneous fractional dimension, $${\rm{S}}{{\rm{d}}_f}({\cal C})$$ , of $${\cal C}$$ is min{h(V): h is a simultaneous resolving function of $${\cal C}$$ }. In this paper, we initiate the study of the simultaneous fractional dimension of a graph family. We obtain $${\max _{1 \le i \le k}}\{{\dim _f}({G_i})\} \le {\rm{S}}{{\rm{d}}_f}({\cal C}) \le \min \{\sum\nolimits_{i = 1}^k {{{\dim}_f}({G_i}),{{|V|} \over 2}} $$ , where both bounds are sharp. We characterize $${\cal C}$$ satisfying $${\rm{S}}{{\rm{d}}_f}({\cal C}) = 1$$ , examine $${\cal C}$$ satisfying $${\rm{S}}{{\rm{d}}_f}({\cal C}) = {{|V|} \over 2}$$ , and determine $${\rm{S}}{{\rm{d}}_f}({\cal C})$$ when $${\cal C}$$ is a family of vertex-transitive graphs. We also obtain some results on the simultaneous fractional dimension of a graph and its complement.