If-then-else inside TikZ graph?

\ifthenelse is "normal" LaTeX code. Therefore you can not use this command inside a TikZ path specification. But since the node text is put in a normal TeX box you can use \ifthenelse inside the node text. So you can try

\documentclass[tikz,margin=5mm]{standalone}
\usetikzlibrary{graphs,graphs.standard,quotes}

\usepackage{ifthen}

\begin{document}
\begin{tikzpicture}[
  vertex/.style={circle,fill=blue!15,draw,minimum size=17pt,inner sep=0pt}
]
%See TikZ documentation! (Section 19, graphs)
\graph[circular placement, radius=4cm, group polar shift=(360/5:0),
         nodes={circle,draw,vertex}] {
    \foreach \x in {0,...,4} {
      \foreach \y in {\x,...,4} {
        \x --["\ifthenelse{\x=3 \OR \y=3 \OR \x=\y}{}{\x\y}",sloped] \y;
  }}};
\end{tikzpicture}
\end{document}

Or you can position the nodes inside the \graph and connect them outside. Then you can use \ifthenelse outside of a path specification. Note that the complete \draw paths are in the arguments of \ifthenelse.

\documentclass[tikz,margin=5mm]{standalone}
\usetikzlibrary{graphs,graphs.standard,quotes}

\usepackage{ifthen}

\begin{document}
\begin{tikzpicture}[
  vertex/.style={circle,fill=blue!15,draw,minimum size=17pt,inner sep=0pt}
]
%See TikZ documentation! (Section 19, graphs)
\graph[circular placement, radius=4cm, group polar shift=(360/5:0),
         nodes={circle,draw,vertex}] {\foreach \x in {0,...,4} \x;
};
\foreach \x in {0,...,4} {
  \foreach \y in {\x,...,4} {
    \ifthenelse{\x=3 \OR \y=3 \OR \x=\y}{\draw(\x)--(\y);}{\draw(\x)--node[auto,sloped]{\x\y}(\y);}
}}
\end{tikzpicture}
\end{document}

The result is the same as above.

---

In the pgfmanual you can also find \pgfextra as a possibilty to suspend temporarly the path construction and do some other things like \ifthenelse. Here is a simple example where the filling color depends on the counter:

\documentclass[tikz,margin=5mm]{standalone}
\newcommand\mycolor{blue!15}
\usepackage{ifthen}
\begin{document}
    \begin{tikzpicture}[
        vertex/.style={circle,draw,minimum size=17pt,inner sep=0pt}
      ]
  \foreach \i in {0,...,4}{
    \path(0,0)\pgfextra{\ifthenelse{\i=2}{\def\mycolor{red!15}}{}}
      (90+\i*72:1)node[vertex,fill=\mycolor]{\i};}
\end{tikzpicture}
\end{document}

This results in

But AFAIK this does not work inside a \graph command.

---

Update:

Solutions without package \ifthen based on a suggestion of @marmot in a comment:

\documentclass[tikz,margin=5mm]{standalone}
\usetikzlibrary{graphs,graphs.standard,quotes}

\begin{document}
\begin{tikzpicture}[
  vertex/.style={circle,fill=blue!15,draw,minimum size=17pt,inner sep=0pt}
]
%See TikZ documentation! (Section 19, graphs)
\graph[circular placement, radius=4cm, group polar shift=(360/5:0),
         nodes={circle,draw,vertex}] {
    \foreach \x in {0,...,4} {
      \foreach \y 
        [evaluate=\y as \z using {int(ifthenelse(\x==3 || \y==3 || \x==\y,1,0))}]
        in {\x,...,4} {
        \x --["\ifnum \z=0 {\x\y}\fi",sloped] \y;
  }}};
\end{tikzpicture}
\end{document}

or

\documentclass[tikz,margin=5mm]{standalone}
\usetikzlibrary{graphs,graphs.standard,quotes}

\begin{document}
\begin{tikzpicture}[
  vertex/.style={circle,fill=blue!15,draw,minimum size=17pt,inner sep=0pt}
]
%See TikZ documentation! (Section 19, graphs)
\graph[circular placement, radius=4cm, group polar shift=(360/5:0),
         nodes={circle,draw,vertex}] {\foreach \x in {0,...,4} \x;
};
\foreach \x in {0,...,4} {
  \foreach \y 
    [evaluate=\y as \z using {int(ifthenelse(\x==3 || \y==3 || \x==\y,1,0))}]
    in {\x,...,4} {
      \ifnum \z=1
        \draw(\x)--(\y);
      \else 
        \draw(\x)--node[auto,sloped]{\x\y}(\y);
      \fi
}}
\end{tikzpicture}
\end{document}

The result is the same as in the first picture.

You can pre-compute an array of points using \ifthenelse, then use them in the \foreach. I have no idea what you are trying to accomplish with the second example.

\documentclass{article}
\usepackage{mathptmx}
\usepackage{tikz}
\usepackage{verbatim}
\usetikzlibrary{arrows,shapes,graphs,graphs.standard,quotes}
\usepackage{calc}% http://ctan.org/pkg/calc
\usepackage{ifthen}

\begin{document}
\begin{tikzpicture}

\def\xset{1,2,3,5}% sort of obvious

\def\xset{\strut}% doing it the hard way
\foreach \x in {1,...,5}
  {\ifthenelse{\NOT 4 = \x \AND \NOT 7 = \x}
  {\if\xset\strut\relax\global\edef\xset{\x}\else\global\edef\xset{\xset,\x}\fi}
  {}};

\foreach \x in {1,...,5}
{ \node[circle,draw] (\x 1) at (0,\x) {\x };
    \node[circle,draw] (\x 2) at (2,\x) {\x };
}
\foreach \x in \xset {\draw (\x 1) -- (\x 2);};

\end{tikzpicture}
\end{document}