\documentclass[tikz]{standalone}

\usepackage{tikz}

\begin{document}
\usetikzlibrary{automata,positioning}

\iffalse
  $S = $ TCATCATT
The suffix language of $S$ is $\{TCATCATT, CATCATT, ATCATT, TCATT, CATT, ATT, AT, T\}$.
  \fi
\begin{tikzpicture}[shorten >=1pt,node distance=2cm,on grid, auto]
  \node[state, initial] (q_0) {$q_{0}$};
  \node[state,right=of q_0,accepting] (q_1) {$q_{1}$};
  \node[state,right=of q_1] (q_2) {$q_{2}$};
  \node[state,right=of q_2] (q_3) {$q_{3}$};
   \node[state,right=of q_3] (q_4) {$q_{4}$};
  \node[state,right=of q_4] (q_5) {$q_{5}$};
  \node[state,right=of q_5] (q_6) {$q_{6}$};
  \node[state,right=of q_6] (q_7) {$q_{7}$};
  \node[state,right=of q_7,accepting] (q_8) {$q_{8}$};

  \path[->] (q_0) edge node {T} (q_1)
  (q_1) edge node {C} (q_2)
  (q_2) edge node {A} (q_3)
  (q_3) edge node {T} (q_4)
  (q_4) edge node {C} (q_5)
  (q_5) edge node {A} (q_6)
  (q_6) edge node {T} (q_7)
  (q_7) edge node {T} (q_8);

  % CATCATT
  \path[->] (q_0) edge[bend right=40] node {C} (q_2);

  % ATCATT
  \path[->] (q_0) edge[bend right=50] node {A} (q_3);

  % TCATT
  \path[->] (q_4) edge[bend right=30] node {T} (q_8);

  % CATT

  % ATT
  %\path[->] (q_0) edge[bend right=50] node {A} (q_6);

  % TT
  \path[->] (q_1) edge[bend left=30] node {T} (q_8);

\end{tikzpicture}
\end{document}