Package TikZ lends itself very well to drawing graphs.

This is a small example (requires TikZ 3.0+):

\documentclass{standalone}

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

\begin{document}
  \begin{tikzpicture}[auto,vertex/.style={draw,circle}]
    \node[vertex] (a) {A};
    \node[vertex,right=1cm of a] (b) {B};
    \node[vertex,below right=1cm and 0.5cm of a] (c) {C};
    
    \path[-{Stealth[]}]
      (a) edge node {1} (b) 
      (a) edge (c)
      (c) edge (b);
  \end{tikzpicture}
\end{document}

You can create arbitrarily complex graphs; beware lengthy code, though. Recall that there is \foreach and take note of all the positioning and styling options (cf. TikZ manual, section 13 to 17).

TikZ provides syntax similar to DOT which you can use to tighten up your graph drawing code considerably.

\documentclass{standalone}

\usepackage{tikz}
\usetikzlibrary{graphs,quotes,arrows.meta}

\begin{document}
  \begin{tikzpicture}
    \graph[nodes={draw,circle},edges={-{Stealth[]}}] {
      A -> ["1"] B, 
      A -> C,
      C -> B
    };
  \end{tikzpicture}
\end{document}

As you can see, you trade fine-grained control for easier syntax. The graphs library really shines when you specify more complicated graphs:

\documentclass{standalone}

\usepackage{tikz}
\usetikzlibrary{graphs,graphs.standard}

\begin{document}
  \begin{tikzpicture}
    \graph {
      A -> { subgraph I_n [V= {B,C,D}] } -> E 
    };
  \end{tikzpicture}
\end{document}

There are many more options and pre-defined graphs; see section 19 of the TikZ manual.

TikZ implements several algorithms for automatic graph layouts (requires LuaLaTeX).

\documentclass{article}

\usepackage{tikz}
\usetikzlibrary{graphs,graphdrawing,quotes}
\usegdlibrary{force}

\begin{document}
  \begin{tikzpicture}
    \graph[spring layout] {
      A -> ["1"] B, 
      A -> {C, D},
      C -> {B, D},
    };
  \end{tikzpicture}
\end{document}

There are several algorithms and many options to influence them. See part IV of the TikZ manual for details.

Suppose the following matrix is the transition probability matrix associated with a Markov chain.

    0.5   0.2  0.3
P=  0.0   0.1  0.9
    0.0   0.0  1.0

In order to study the nature of the states of a Markov chain, a state transition diagram of the Markov chain is drawn.

\documentclass[12pt,a4paper]{article}
\usepackage{tikz}
\usetikzlibrary{shapes,arrows,positioning}
\begin{tikzpicture}[->,>=stealth',shorten >=2pt, line width=3pt, 
                                  node distance=2cm, style ={minimum size=20mm}]
\tikzstyle{every node}=[font=\huge]
\node [circle, draw] (a) {1};
\path  (a) edge [loop above] (a);
\node [circle, draw] (b) [right=of a] {2};
\path  (b) edge [loop above] (b);
\draw[->] (a) -- (b);
\node [circle, draw] (c) [below=of a] {3};
\path  (c) edge [loop below] (c);
\draw[->] (a) -- (c);
\draw[->] (b) -- (c);
\end{tikzpicture}