how to obtain state space diagram and state space model for transfer function How do we obtain a state-space realization and a block diagram of a given transfer function?
Consider the transfer function
$$\frac{C(s)}{R(s)}=\frac{5s}{3s^{2}+3s+1}$$
Steps for solution are
$$\frac{C(s)}{R(s)}=\frac{5s}{3s^{2}+3s+1}\frac{Q(s)}{Q(s)}\\$$
$$C(s)=5(s^{-1})Q(s)\\$$
$$\Rightarrow R(s)=(3+3s^{-1}+s^{-2})Q(s)$$
$$\Rightarrow R(s)=3Q(s)+3(s^{-1})Q(s)+s^{-2}Q(s)$$
$$\Rightarrow 3Q(s)=R(s)-3(s^{-1})Q(s)-s^{-2}Q(s)$$
$$\Rightarrow Q(s) = \frac13R(s)-s^{-1}Q(s)-\frac13s^{-2}Q(s)$$
$$\Rightarrow Q(s) = \frac13R(s)-Q(s)\left[s^{-1}+\frac13s^{-2}\right]$$
the graph which is plotted in the book is of last equation of above solution.
I do not know how to post the graph here on Stack Exchange, but what I want to understand is: 

How is the graph of this equation plotted?

 A: Only some parts of your question make sense. In particular, given the transfer function 
$$
H(s) = \frac{5s}{3s^2+3s+1} 
$$
you invert the Laplace transforms to obtain the ode
$$3y''(t) + 3y'(t) + y = 5u'(t).$$
Define $X_1 = y$, $X_2 = y'$, $U_1 = u$, $U_2 = u'$. The state equations may therefore be written as
$$
\begin{aligned}
X_1' &= X_2\\
X_2'  &= -X_2 - \frac{1}{3}X_1 + \frac{5}{3}U_2\\
\end{aligned}
$$
and thus
$$
\begin{bmatrix}
X_1\\
X_2\\
\end{bmatrix}'
= \begin{bmatrix}
0 & 1\\
-\frac{1}{3} & -1
\end{bmatrix}
\begin{bmatrix}
X_1\\
X_2\\
\end{bmatrix}
+
\begin{bmatrix}
0 & 0\\
0 &
\frac{5}{3}
\end{bmatrix}
\begin{bmatrix}
U_1\\
U_2\\
\end{bmatrix},
$$
corresponding to the partial 
realization
$$
A = \begin{bmatrix}
0 & 1\\
-\frac{1}{3} & -1
\end{bmatrix}
,\ B = \begin{bmatrix}
0 & 0\\
0 &
\frac{5}{3}
\end{bmatrix}.
$$
This is the only interpretation I can think of for "state space model". The only thing I can imagine is meant by "state variable diagram" (other than a block diagram which there is no "plotting" involved in making) is the $X_1$ vs. $X_2$ contour plot which you get by plotting the vector field of the RHS minus the controller parametrically. I have no idea what the calculation you show is trying to do and without any reference to where you got it from have no idea where to start with that. 
