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 C={G1,G2,…,Gk}\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$${\cal C} = \{{G_1},{G_2}, \ldots ,{G_k}\} $$\end{document} 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\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$${\cal C}$$\end{document} if h(RG{x, y}) ≥ 1 for any distinct vertices x, y ∈ V and for every graph G∈C\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$G \in {\cal C}$$\end{document}. The simultaneous fractional dimension, Sdf(C)\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$${\rm{S}}{{\rm{d}}_f}({\cal C})$$\end{document}, of C\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$${\cal C}$$\end{document} is min{h(V): h is a simultaneous resolving function of C\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$${\cal C}$$\end{document}}. In this paper, we initiate the study of the simultaneous fractional dimension of a graph family. We obtain max1≤i≤k{dimf(Gi)}≤Sdf(C)≤min{∑i=1kdimf(Gi),|V|2\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$${\max _{1 \le i \le k}}\{{\dim _f}({G_i})\} \le {\rm{S}}{{\rm{d}}_f}({\cal C}) \le \min \{\sum\nolimits_{i = 1}^k {{{\dim}_f}({G_i}),{{|V|} \over 2}} $$\end{document}, where both bounds are sharp. We characterize C\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$${\cal C}$$\end{document} satisfying Sdf(C)=1\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$${\rm{S}}{{\rm{d}}_f}({\cal C}) = 1$$\end{document}, examine C\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$${\cal C}$$\end{document} satisfying Sdf(C)=|V|2\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$${\rm{S}}{{\rm{d}}_f}({\cal C}) = {{|V|} \over 2}$$\end{document}, and determine Sdf(C)\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$${\rm{S}}{{\rm{d}}_f}({\cal C})$$\end{document} when C\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$${\cal C}$$\end{document} is a family of vertex-transitive graphs. We also obtain some results on the simultaneous fractional dimension of a graph and its complement.
