## Random Walk on Graph

a) Let K5 be the graph with vertices {1,2,3,4,5} and edges {12,23,34,45,15} (a cycle). Consider the random walk on K5.
Is this the correct Transition matrix?

$

\begin{pmatrix}
0 & 0.5 & 0 & 0 & 0.5\\
0.5 & 0 & 0. 5 & 0 & 0\\
0 & 0.5 & 0 & 0.5 & 0\\
0 & 0 & 0.5 & 0 & 0.5\\
0.5 & 0 & 0 & 0.5 & 0\\
\end{pmatrix}
$

i) How do you know that this has a unique equilibrium distribution?
ii) How do you know that it has a limiting distribution?
iii) Without solving any equations write down the equilibrium distribution.

thanks for any help.