Let $\mathcal{G}=(V,E,\ell)$ be a finite labelled directed graph over a finite alphabet $\mathcal{A}$, where $\ell:E\to\mathcal{A}$ is the edge-labelling map. Assume that $\mathcal{G}$ is right-resolving: for every $v\in V$ and every $a\in\mathcal{A}$, there is at most one edge $e\in E$ with initial vertex $v$ and label $\ell(e)=a$.

Let $Y\subseteq \mathcal{A}^{\mathbb{Z}}$ be the nonempty sofic shift presented by $\mathcal{G}$, meaning that $Y$ consists exactly of the bi-infinite label sequences read along bi-infinite paths in $\mathcal{G}$. Suppose that $\mathcal{G}$ is essential in the sense that every vertex of $\mathcal{G}$ lies on a bi-infinite path whose label sequence belongs to $Y$. Let $M\in \mathbb{N}^{V\times V}$ be the underlying adjacency matrix of $\mathcal{G}$, defined by

\begin{align*} M_{uv}=\#\{e\in E:\text{ the initial vertex of }e\text{ is }u\text{ and the terminal vertex of }e\text{ is }v\}. \end{align*}

Then

\begin{align*} h_{\mathrm{top}}(\sigma|_Y)=\log \rho(M), \end{align*}

where $\rho(M)$ denotes the spectral radius of $M$.

If $\mathcal{G}$ is not essential, let $\mathcal{G}_{\mathrm{ess}}$ be the subgraph spanned by the vertices and edges that occur in some bi-infinite path whose label sequence belongs to $Y$, and let $M_{\mathrm{ess}}$ be its underlying adjacency matrix. Then

\begin{align*} h_{\mathrm{top}}(\sigma|_Y)=\log \rho(M_{\mathrm{ess}}). \end{align*}