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Let t $\in \mathbb{N}$. Then, let $W_t(x,y)$ be the number of walks that begins on x and end on y, with x,y $\in \mathbb{Z}_3$ ( you can think it like a cycle with 3 points). The definition of walk on a graph, for me is a sequence of vertex and edges, and in this particular case, we don't have walks that stay in the same point (lazy ones), because we don't have edges {x,x}...

Then, what is the function $W_t(0,0)$ and $W_t(0,1)$?

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    Look for a two term recurrence – 2017-01-24
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    Matrix view: $$W_t =\begin{pmatrix}0&1&1\\1&0&1\\1&1&0\end{pmatrix}^t$$ – 2017-01-24
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    Now $W_1^2=W_1+2I$... – 2017-01-24
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    I did not understand the matrix. But recurrence i have 3 equation and i'm stuck... – 2017-01-24

1 Answers 1

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Let $a_t$ be the number of ways to start at $0$ and end at $0$ in $t$ steps.

Let $b_t$ be the number of ways o start at $0$ and end at $1$ or $2$.

Then show: $a_{t+1}=b_t, b_{t+1}=2a_t+b_t$.

Combining these, you get that $b_{t+1}=b_t+2b_{t-1}$.

Solve for $b_{t}$.

Finally, $W_{t}(0,0)=a_t, W_{t}(0,1)=\frac{1}{2}b_t$.