The Simultaneous Fractional Dimension of Graph Families

Abstract

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∈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(R_G{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.