Let $k \in \mathbb{N}$ and let $M \in \{0,1\}^{k \times k}$ be a zero-one matrix. Define the one-sided topological Markov chain
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\begin{align*}
\Sigma_M := \{x = (x_n)_{n \in \mathbb{N}} \in \{1,\dots,k\}^{\mathbb{N}} : M_{x_n x_{n+1}} = 1 \text{ for every } n \in \mathbb{N}\}.
\end{align*}
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Assume that $\Sigma_M \neq \varnothing$, and let $\sigma|_{\Sigma_M}: \Sigma_M \to \Sigma_M$ be the left shift restricted to $\Sigma_M$. If $\rho(M)$ denotes the spectral radius of $M$, then $\rho(M) > 0$ and
Moreover, if $M$ is reducible and $M_1,\dots,M_r$ are the irreducible diagonal blocks obtained from the strongly connected components of the directed graph of $M$, then