I assume you're using TikZ and tkz-graph …
You can use the decoration.markings library to set a marking at the middle position.
To the human eye this does not look like the middle, though, because only the tip of the arrow is halfway between (1) and (2).
But you can use arbitrary TikZ code in the with {<code goes here>} part.
I cloned the first solution and changed it so that it does draw a zero length arrow (without a line). To me it looks better than the first solution.
Code
\documentclass{article}
\usepackage{tikz,tkz-graph}
\usetikzlibrary{calc}
\usetikzlibrary{decorations.markings}
\begin{document}
\SetVertexNormal[
Shape = circle,
LineWidth = 2pt
]
\SetUpEdge[
lw = 1.5pt,
color = blue
]
\begin{tikzpicture}
\Vertex[x=0 ,y=0]{1}
\Vertex[x=0 ,y=2]{2}
\tikzset{EdgeStyle/.append style = {straight}}
\tikzset{
EdgeStyle/.style={
postaction=decorate,
decoration={
markings,
mark=at position .5 with {\arrow{>}}
}
}
}
\Edge(1)(2)
\Vertex[x=0 ,y=0]{1}
\Vertex[x=0 ,y=2]{2}
\Vertex[x=1 ,y=0]{1}
\Vertex[x=1 ,y=2]{2}
\tikzset{
EdgeStyle/.style={
postaction=decorate,
decoration={
markings,
mark=at position .5 with {\draw[red,->] (0,0) (0,0);}
}
}
}
\Edge(1)(2)
\Vertex[x=1 ,y=0]{1}
\Vertex[x=1 ,y=2]{2}
\end{tikzpicture}
\end{document}
Output

Closer look

More solutions can be found:
TikZ: How to draw an arrow in the middle of the line?
texdoc --view pgfmanual), specially the chapter "Arrow Tip Library". You are probably looking fortriangle 60. – Qrrbrbirlbel Sep 29 '12 at 18:41