I wanted to plot a flow chart graph which looks similar to this diagram:

that comes from this source.
- Which package would be best for that?
- I don't want to use pstricks
I wanted to plot a flow chart graph which looks similar to this diagram:

that comes from this source.
An example with pgf/tikZ just for your inspiration.
\documentclass{minimal}
\usepackage{tikz}
\usetikzlibrary{positioning,shapes.geometric}
\begin{document}
\begin{tikzpicture}[%
->,
shorten >=2pt,
>=stealth,
node distance=1cm,
noname/.style={%
ellipse,
minimum width=5em,
minimum height=3em,
draw
}
]
\node[noname] (1) {1};
\node[noname] (2) [below=of 1] {2};
\node[noname] (4) [node distance=1cm and 3mm,below left=of 2] {4};
\node[noname] (3) [left=of 4] {3};
\node[noname] (5) [below=of 4] {5};
\node[noname] (6) [node distance=2cm,right=of 5] {6};
\path (1) edge node {} (2)
(2) edge node {} (3)
(2) edge node {} (4)
(2) edge node {} (6)
(3) edge node {} (5)
(4) edge node {} (5)
(5) edge [bend right=20pt] node {} (2);
\end{tikzpicture}
\end{document}
More examples in the according tikZ Example Gallery.

This doesn't need TeX. You can just use graphviz to draw such graph.
Of course, many LaTeX packages like pstricks and tikz can draw these graph, but I think graphviz is still easier to use.
It seems an example copied from main page of graphviz will be useful. With graphviz we don't have to tell computer where the nodes are, only edges are needed.
digraph finite_state_machine {
rankdir=LR;
size="8,5"
node [shape = doublecircle]; LR_0 LR_3 LR_4 LR_8;
node [shape = circle];
LR_0 -> LR_2 [ label = "SS(B)" ];
LR_0 -> LR_1 [ label = "SS(S)" ];
LR_1 -> LR_3 [ label = "S($end)" ];
LR_2 -> LR_6 [ label = "SS(b)" ];
LR_2 -> LR_5 [ label = "SS(a)" ];
LR_2 -> LR_4 [ label = "S(A)" ];
LR_5 -> LR_7 [ label = "S(b)" ];
LR_5 -> LR_5 [ label = "S(a)" ];
LR_6 -> LR_6 [ label = "S(b)" ];
LR_6 -> LR_5 [ label = "S(a)" ];
LR_7 -> LR_8 [ label = "S(b)" ];
LR_7 -> LR_5 [ label = "S(a)" ];
LR_8 -> LR_6 [ label = "S(b)" ];
LR_8 -> LR_5 [ label = "S(a)" ];
}

graphviz is not a part of TeX. Just use \includegraphics to import the graph.
– Leo Liu
Jan 16 '11 at 15:19
dot2tex on the dot-file. It will yield a .tex file containing the tikZ command you're after
– 19h
Mar 19 '16 at 11:46
If you want to draw graph (graph theory) diagram, then PSTricks can do this easily.
See this tutorial.