Fraction of \textwidth as image width

Informally, length calculation are done using factor multiplication as there is no division notation. Since any division can be represented as some multiplication, this shouldn't be a problem. So, .5\textwidth refers to half (1/2) of \textwidth, while 2\wd0 refers to twice the width of box 0. In your case it suffices to use

\includegraphics[width=0.3333\textwidth]{<img>}

You can perform all kinds of calculations using calc or even LaTeX3:

\documentclass{article}

\usepackage{xparse}
\ExplSyntaxOn
\cs_new_eq:NN \calc \fp_eval:n
\ExplSyntaxOff

\begin{document}

\newlength{\mylen}
\setlength{\mylen}{\textwidth}

\verb|\mylen|: \the\mylen

\setlength{\mylen}{\calc{1/3}\textwidth}
\verb|1/3\mylen|: \the\mylen

\end{document}

The LaTeX3 syntax above makes \calc an new control sequence (or macro) that is equivalent to \fp_eval:n - a floating point function that evaluates it's argument using the regular programming arithmetic (like +, -, *, /, ^, ...) taking a single argument.

Compile with lualatex

\documentclass{article}
\usepackage[demo]{graphicx}
\usepackage{luacode}
\begin{document}

\newcommand\w{\directlua{tex.sprint(1/3)}}

This is my graph, it has width of \w and it looks very nice

\includegraphics[width=\w\textwidth]{}

\end{document}