[Tex/LaTex] Parallel edges in Graph using \path

tikz-pgf

I'm trying to get Parallel edges between nodes.
I recognize that very similar questions have been asked before, however;

  1. Those solutions don't seem to have worked on my problem, though
    my search was not exhaustive. Most solutions seem to use \draw, and
    I don't yet have the skill to reconcile \path approach with \draw.
    Further, I couldn't find mention in the 3.0.0 documentation.

  2. I'd like to keep this style consistent with other graphs that I
    have done for a report, as I already have a few examples working. If
    someone knows how to remove the arrow from the loop that would be
    very helpful also. As you can see I have tried using "stealth", but
    haven't had any luck.

My code so far:

\documentclass[14pt,oneside,a4paper]{report}
\usepackage{tikz}
 \usetikzlibrary{arrows,decorations.pathmorphing,backgrounds,positioning,fit,petri}
\begin{document}
\begin{figure}

\sbox{\tempbox}{%
\begin{tikzpicture}[-,>=stealth',shorten >=1pt,auto,node distance=2cm,
  thick,main node/.style={circle,fill=black!20,draw}]

  \node[main node] (3) {$v_3$};
  \node[main node] (1) [below left of=3] {$v_1$};
  \node[main node] (2) [below right of=3] {$v_2$};

  \path[every node/.style={font=\sffamily\small}]
    (1) edge[loop left] node  {$e_{1}$} (1)
    (1) edge node [left] {} (2)
    (1) edge node [left] {} (3)
    (2) edge node [right] {} (3);
    \path [draw, line width=3] (2) |-(3);
\end{tikzpicture}
} 

\begin{subfigure}{.5\textwidth}
\centering
\usebox{\tempbox}
\caption{A non-simple Graph}
\end{subfigure}%
\begin{subfigure}{.5\textwidth}
\centering
\vbox to\ht\tempbox{
    \vfill
\begin{math}
A_{G} = \left(
\begin{array}{ccc}
1 & 1 & 1 \\
1 & 0 & 2 \\
1 & 2 & 0 

\end{array}
\right)
\end{math}
\vfill
}
\caption{Matrix Corresponding to the Graph }
\end{subfigure}%
\caption{A graph's diagram and it's adjacency matrix representation.}
\end{figure}
\end{document}

The graphics output

The top Graph is of the style I'm aiming for.
The Graph below it is what the code posted creates.
The sharp-angled edge is the one which i'm wishing to modify to be smoother.
I've tried using:

(2) edge node [bend left = 15] {} (3);

However that didn't work.
NB: I really don't know what I'm doing. I included the Matrix as I'm working in a figure environment, if that is meaningful.

Best Answer

Edit: In the first attempt I miss understand question, so I add arrows instead to remove it from loop. Hot to do this is described in TikZ manual on page 749 (version 3.0.1a)

Something like this?

enter image description here

\documentclass{article}
\usepackage{subcaption}

\usepackage{tikz}
    \usetikzlibrary{arrows,positioning}

\begin{document}
\begin{figure}

\newsavebox\tempbox
\sbox{\tempbox}{%
\begin{tikzpicture}[
    thick, shorten >=1pt, 
    auto,
    node distance=2cm,
  main node/.style={circle, draw, fill=black!20}
                    ]
\node[main node] (3) {$v_3$};
\node[main node] (1) [below left =of 3] {$v_1$};% <--- se proper use of positioning 
\node[main node] (2) [below right=of 3] {$v_2$};

  \path[every loop/.style={}]
    (1) edge[loop left,"$e_{1}$"] (1)% <-- for edge labels I use quotes library

    (1) edge (2)% <-- nodes are removed since they are not used
    (1) edge (3)
    (2) edge (3)
    (3) edge [red, bend left] (2); % <-- edge is now smooth
\end{tikzpicture}
}

\begin{subfigure}{.5\textwidth}
\centering
\usebox{\tempbox}
\caption{A non-simple Graph}
\end{subfigure}%
\begin{subfigure}{.5\textwidth}
\centering
\vbox to\ht\tempbox{
    \vfill
\begin{math}
A_{G} = \left(
\begin{array}{ccc}
1 & 1 & 1 \\
1 & 0 & 2 \\
1 & 2 & 0
\end{array}
\right)
\end{math}
\vfill
}
\caption{Matrix Corresponding to the Graph }
\end{subfigure}%
\caption{A graph's diagram and it's adjacency matrix representation.}
\end{figure}
    \end{document}

To code I add comments to see, what I change in your MWE

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