# Let G be the graph with vertices v_{1}, v_{2} and v_{3} and the matrix begi

Let G be the graph with vertices ${v}_{1},{v}_{2}$ and ${v}_{3}$ and the matrix $\left[\begin{array}{ccc}1& 1& 2\\ 1& 0& 1\\ 2& 2& 0\end{array}\right]$
To find the number of walks of from ${v}_{1}$ to ${v}_{3}$ we need to find matrix ${A}^{2}$

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Step 1
${A}^{2}=\left[\begin{array}{ccc}1& 1& 2\\ 1& 0& 1\\ 2& 2& 0\end{array}\right]×\left[\begin{array}{ccc}1& 1& 2\\ 1& 0& 1\\ 2& 2& 0\end{array}\right]$
${A}^{2}=\left[\begin{array}{ccc}6& 5& 3\\ 3& 3& 2\\ 4& 2& 6\end{array}\right]$
The $i{j}^{th}$ element in matrix ${A}^{2}$ represents the number of walks from ${v}_{i}$ to ${v}_{j}$ of length 2.
We can see from matrix ${A}^{2}$ that the element ${a}_{13}$ is 3. This implies that there are 3 walks of length 2 from ${v}_{1}$ to ${v}_{3}$