I have the following:

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}

now I want to adjust the spacing in the caption in the sense that i want to "push it into the middle", but I don't know how to proceed.

EDIT: similar question as above but the caption looks awful in this case, any fixes to stretch it out:

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[1.1FBwidth]{%

 caption{An example of a vertex set $V = lbrace 1, 2, 3, 4, 5, 6rbrace$ 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}

is there a way to stretch this out into 2 lines at most?

If I understand you right, you can use the following two lines in your preamble

usepackage{caption} % <================================================

captionsetup{width=0.8textwidth} % <==================================

to let the caption use only 80% of textwidth. See the documentation of package caption for more possibilitys to manipulate the layout of captions with typing texdoc caption on ypur console/terminal.

The complete code

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=0.8textwidth} % <==================================



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}

gives you the result:

EDIT:

With your second example (after commenting ffigbox, see markings <====== in code)

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=0.8textwidth} % <==================================



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[1.1FBwidth]{% <==============================================

 caption{An example of a vertex set $V = lbrace 1, 2, 3, 4, 5, 6rbrace$ 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}

you get the resulting second figure/page:

If I've well understood what you want, this is easy with the ffigbox command from floatrow, which gives full control on the caption width:

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-1pmod 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[1.1FBwidth]{%

 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}