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EDITAR:

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EDITAR:

Eu tenho o seguinte:

\documentclass[11pt, a4paper]{report}
\usepackage{bm}
\usepackage{amsfonts, graphicx, verbatim, amsmath,amssymb, amsthm}
\usepackage{color}
\usepackage{array}
\usepackage{setspace}% if you must (for double spacing thesis)
\usepackage{fancyhdr}
\usepackage{enumitem}
\usepackage{tikz}
\usepackage{parskip}
\usepackage{lipsum}
    \newtheorem{theorem}{Theorem}[section]
\newtheorem{example}[theorem]{Example}   

 \begin{document}
    \begin{example} 
    Consider a random walk on the $n$-cycle. Let $\Omega = \mathbb{Z}_n = \lbrace 0, 1, 2, \cdots, n-1 \rbrace$ be the set of remainders modulo $n$. Also consider the transition matrix:
    \[
    P(x,y) = 
    \begin{cases} 
    \frac{1}{2} & \text{if } y=x+1\;\; (mod\;n)\\
    \frac{1}{2} & \text{if } y=x-1\;\; (mod\;n)\\
    0 & \text{otherwise}
    \end{cases}
    \]

    The associated Markov chain $X_t$ is called a random walk on the $n$-cycle. The sates can be visualised as equally spaced nodes arranged in a circle(see figure 1.1)
    \end{example}

    \begin{figure}[htbp]
        \centering
        \begin{tikzpicture}
        \foreach \i in {90,54,...,-234} {
            \draw[ultra thick] (\i:2)--({\i-36}:2);
        }
        \foreach \i in {90,18,...,-198} {
            \draw[fill=black] (\i:2) circle (1.25mm);
        }
        \foreach \i in {54,-18,...,-234} {
            \draw[fill=white] (\i:2) circle (1.25mm);
        }
        \begin{scope}[xshift=5cm]
        \foreach \i in {90,50,...,-230} {
            \draw[ultra thick] (\i:2)--({\i-40}:2);
            \draw[fill=black] (\i:2) circle (1.25mm);
        }
        \end{scope}
        \end{tikzpicture}
        \caption{Random walk on $\mathbb{Z}_10$ is periodic, since every step
    goes from an even state to an odd state, or vice-versa. Random
    walk on $\mathbb{Z}_9$ is aperiodic.}
        \label{my:figure}
    \end{figure}
    \end{document}

agora quero ajustar o espaçamento da legenda no sentido de que quero "empurrá-la para o meio", mas não sei como proceder.

EDITAR: pergunta semelhante à acima, mas a legenda parece horrível neste caso, qualquer correção para estendê-la:

\begin{example} Consider the graph $G$ following shown in figure 1.2. The transition matrix of a simple random walk $G$ is 

\begin{equation*}
P =
\begin{bmatrix}[1.25]
    0 & \frac{1}{3} & \frac{1}{3} & \frac{1}{3} & 0 & 0 \\
    \frac{1}{4} & 0 & \frac{1}{4} & 0 & \frac{1}{4} & \frac{1}{4} \\
    \frac{1}{4} & \frac{1}{4} & 0 & \frac{1}{4} & \frac{1}{4} & 0 \\
    \frac{1}{3} & 0 & \frac{1}{3} & 0 & \frac{1}{3} & 0 \\
    0 & \frac{1}{4} & \frac{1}{4} & \frac{1}{4} & 0 & \frac{1}{4} \\
    0 & \frac{1}{2} & 0 & 0 & \frac{1}{2} & 0 \\
\end{bmatrix}
\end{equation*}

\begin{figure}[htbp]
 \centering
 \ffigbox[1.1\FBwidth]{%
 \caption{An example of a vertex set $V = \lbrace 1, 2, 3, 4, 5, 6\rbrace$ with $10$ edges.}
 \label{my:figure}}%
 {\begin{tikzpicture}[bn/.style={circle,fill,draw,text=white,font=\sffamily,minimum
size=1mm},every node/.append style={bn}]
 \path node (1) {1} -- ++ (50:2.5) node (2) {2} -- ++(-95:1.75) node (3) {3}
 -- ++(-85:1.75) node (4) {4} -- ++(40:2.75) node (5) {5}
 -- ++ (0,1.75) node (6) {6} ;
 \draw[thick] (1)--(2)--(6)--(5)--(4)--(1)--(3)--(5)--(2)--(3)--(4);
\end{tikzpicture}}%
\end{figure}
\end{example}

existe uma maneira de esticar isso em no máximo duas linhas?

Responder1

Se bem entendi, você pode usar as duas linhas a seguir em seu preâmbulo

\usepackage{caption} % <================================================
\captionsetup{width=0.8\textwidth} % <==================================

para permitir que a legenda use apenas 80% da largura do texto. Veja a documentação do pacote captionpara mais possibilidades de manipular o layout das legendas digitando texdoc captionno seu console/terminal.

O código completo

\documentclass[11pt, a4paper]{report}
\usepackage{bm}
\usepackage{amsfonts, graphicx, verbatim, amsmath,amssymb, amsthm}
\usepackage{color}
\usepackage{array}
\usepackage{setspace}% if you must (for double spacing thesis)
\usepackage{fancyhdr}
\usepackage{enumitem}
\usepackage{tikz}
\usepackage{parskip}
\usepackage{lipsum}

\usepackage{caption} % <================================================
\captionsetup{width=0.8\textwidth} % <==================================

\newtheorem{theorem}{Theorem}[section]
\newtheorem{example}[theorem]{Example}   


\begin{document}

\begin{example} 
Consider a random walk on the $n$-cycle. 
Let $\Omega = \mathbb{Z}_n = \lbrace 0, 1, 2, \cdots, n-1 \rbrace$ be 
the set of remainders modulo $n$. Also consider the transition matrix:
\[
  P(x,y) = 
  \begin{cases} 
    \frac{1}{2} & \text{if } y=x+1\;\; (mod\;n)\\
    \frac{1}{2} & \text{if } y=x-1\;\; (mod\;n)\\
    0           & \text{otherwise}
  \end{cases}
\]

The associated Markov chain $X_t$ is called a random walk on the 
$n$-cycle. The sates can be visualised as equally spaced nodes arranged 
in a circle (see figure~\ref{my:figure}). % <==========================
\end{example}

\begin{figure}[htbp]
  \centering
  \begin{tikzpicture}
        \foreach \i in {90,54,...,-234} {
            \draw[ultra thick] (\i:2)--({\i-36}:2);
        }
        \foreach \i in {90,18,...,-198} {
            \draw[fill=black] (\i:2) circle (1.25mm);
        }
        \foreach \i in {54,-18,...,-234} {
            \draw[fill=white] (\i:2) circle (1.25mm);
        }
        \begin{scope}[xshift=5cm]
        \foreach \i in {90,50,...,-230} {
            \draw[ultra thick] (\i:2)--({\i-40}:2);
            \draw[fill=black] (\i:2) circle (1.25mm);
        }
        \end{scope}
  \end{tikzpicture}
  \caption{Random walk on $\mathbb{Z}_10$ is periodic, since every step
    goes from an even state to an odd state, or vice-versa. Random
    walk on $\mathbb{Z}_9$ is aperiodic.}
  \label{my:figure}
\end{figure}
Text after the figure.
\end{document}

te dá o resultado:

pdf resultante

EDITAR:

Com seu segundo exemplo (depois de comentar \ffigbox, veja as marcações <======no código)

\documentclass[11pt, a4paper]{report}
\usepackage{bm}
\usepackage{amsfonts, graphicx, verbatim, amsmath,amssymb, amsthm}
\usepackage{color}
\usepackage{array}
\usepackage{setspace}% if you must (for double spacing thesis)
\usepackage{fancyhdr}
\usepackage{enumitem}
\usepackage{tikz}
\usepackage{parskip}
\usepackage{lipsum}

\usepackage{floatrow}

\usepackage{caption} % <================================================
\captionsetup{width=0.8\textwidth} % <==================================

\newtheorem{theorem}{Theorem}[section]
\newtheorem{example}[theorem]{Example}   


\begin{document}

\begin{example} 
Consider a random walk on the $n$-cycle. 
Let $\Omega = \mathbb{Z}_n = \lbrace 0, 1, 2, \cdots, n-1 \rbrace$ be 
the set of remainders modulo $n$. Also consider the transition matrix:
\[
  P(x,y) = 
  \begin{cases} 
    \frac{1}{2} & \text{if } y=x+1\;\; (mod\;n)\\
    \frac{1}{2} & \text{if } y=x-1\;\; (mod\;n)\\
    0           & \text{otherwise}
  \end{cases}
\]

The associated Markov chain $X_t$ is called a random walk on the 
$n$-cycle. The sates can be visualised as equally spaced nodes arranged 
in a circle (see figure~\ref{my:figure}).
\end{example}

\begin{figure}[htbp]
  \centering
  \begin{tikzpicture}
        \foreach \i in {90,54,...,-234} {
            \draw[ultra thick] (\i:2)--({\i-36}:2);
        }
        \foreach \i in {90,18,...,-198} {
            \draw[fill=black] (\i:2) circle (1.25mm);
        }
        \foreach \i in {54,-18,...,-234} {
            \draw[fill=white] (\i:2) circle (1.25mm);
        }
        \begin{scope}[xshift=5cm]
        \foreach \i in {90,50,...,-230} {
            \draw[ultra thick] (\i:2)--({\i-40}:2);
            \draw[fill=black] (\i:2) circle (1.25mm);
        }
        \end{scope}
  \end{tikzpicture}
  \caption{Random walk on $\mathbb{Z}_10$ is periodic, since every step
    goes from an even state to an odd state, or vice-versa. Random
    walk on $\mathbb{Z}_9$ is aperiodic.}
  \label{my:figure}
\end{figure}
Text after the figure.

\clearpage
\begin{example} Consider the graph $G$ following shown in figure 1.2. The transition matrix of a simple random walk $G$ is 

\begin{equation*}
P =
\begin{bmatrix}[1.25]
    0 & \frac{1}{3} & \frac{1}{3} & \frac{1}{3} & 0 & 0 \\
    \frac{1}{4} & 0 & \frac{1}{4} & 0 & \frac{1}{4} & \frac{1}{4} \\
    \frac{1}{4} & \frac{1}{4} & 0 & \frac{1}{4} & \frac{1}{4} & 0 \\
    \frac{1}{3} & 0 & \frac{1}{3} & 0 & \frac{1}{3} & 0 \\
    0 & \frac{1}{4} & \frac{1}{4} & \frac{1}{4} & 0 & \frac{1}{4} \\
    0 & \frac{1}{2} & 0 & 0 & \frac{1}{2} & 0 \\
\end{bmatrix}
\end{equation*}

\begin{figure}[htbp]
 \centering
%\ffigbox[1.1\FBwidth]{% <==============================================
 \caption{An example of a vertex set $V = \lbrace 1, 2, 3, 4, 5, 6\rbrace$ with $10$ edges.}
 \label{my:figure}%
%}% <===================================================================
 {\begin{tikzpicture}[bn/.style={circle,fill,draw,text=white,font=\sffamily,minimum
size=1mm},every node/.append style={bn}]
 \path node (1) {1} -- ++ (50:2.5) node (2) {2} -- ++(-95:1.75) node (3) {3}
 -- ++(-85:1.75) node (4) {4} -- ++(40:2.75) node (5) {5}
 -- ++ (0,1.75) node (6) {6} ;
 \draw[thick] (1)--(2)--(6)--(5)--(4)--(1)--(3)--(5)--(2)--(3)--(4);
\end{tikzpicture}}%
\end{figure}
\end{example}
\end{document}

você obtém a segunda figura/página resultante:

segunda página

Responder2

Se entendi bem o que você quer, isso é fácil com o \ffigboxcomando from floatrow, que dá controle total sobre a largura da legenda:

\documentclass[11pt, a4paper]{report}
\usepackage{bm}
\usepackage{graphicx, verbatim, amsmath,amssymb, amsthm}
\usepackage{color}
\usepackage{array}
\usepackage{setspace}% if you must (for double spacing thesis)
\usepackage{fancyhdr}
\usepackage{enumitem}
\usepackage{tikz}
\usepackage{parskip}
\usepackage{lipsum}
    \newtheorem{theorem}{Theorem}[section]
\newtheorem{example}[theorem]{Example}
\usepackage{floatrow}

 \begin{document}

    \begin{example}
    Consider a random walk on the $n$-cycle. Let $\Omega = \mathbb{Z}_n = \lbrace 0, 1, 2, \cdots, n-1 \rbrace$ be the set of remainders modulo $n$. Also consider the transition matrix:
    \[
    P(x,y) =
    \begin{cases}
    \frac{1}{2} & \text{if } y=x+1 \pmod n\\
    \frac{1}{2} & \text{if } y=x-1\pmod n\\
    0 & \text{otherwise}
    \end{cases}
    \]

    The associated Markov chain $X_t$ is called a random walk on the $n$-cycle. The states can be visualised as equally spaced nodes arranged in a circle(see figure 1.1)
    \end{example}

 \begin{figure}[htbp]
 \centering
 \ffigbox[1.1\FBwidth]{%
 \caption{Random walk on $\mathbb{Z}_10$ is periodic, since every step
 goes from an even state to an odd state, or vice-versa. Random
 walk on $\mathbb{Z}_9$ is aperiodic.}
 \label{my:figure}}%
 {\begin{tikzpicture}
 \foreach \i in {90,54,...,-234} {
 \draw[ultra thick] (\i:2)--({\i-36}:2);
 }
 \foreach \i in {90,18,...,-198} {
 \draw[fill=black] (\i:2) circle (1.25mm);
 }
 \foreach \i in {54,-18,...,-234} {
 \draw[fill=white] (\i:2) circle (1.25mm);
 }
 \begin{scope}[xshift=5cm]
 \foreach \i in {90,50,...,-230} {
 \draw[ultra thick] (\i:2)--({\i-40}:2);
 \draw[fill=black] (\i:2) circle (1.25mm);
 }
 \end{scope}
 \end{tikzpicture}}%
 \end{figure}

\end{document}

insira a descrição da imagem aqui

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