Coxeter graph of the group $\mathbb{Z}_2 \times \mathbb{Z}_2 \times \dots \times \mathbb{Z}_2$

Question

I am reading the first chapter of Combinatorics of Coxeter Groups by A.Björner and F.Brenti. In the first example they say that the graph with $n$ isolated vertices (no edges) is the Coxeter graph of the group $\mathbb{Z}_2 \times \mathbb{Z}_2 \times \dots \times \mathbb{Z}_2$ of order $2^n$. I just want to make sure that I understand this statement correctly. Let us consider an example. Let $S=\{1,2,3\}$ and the corresponding matrix $m: S \times S \rightarrow \{1,2,\dots,\infty \}$ be \begin{pmatrix} 1 & 2 & 2\\ 2 & 1 & 2\\ 2 & 2 & 1 \end{pmatrix} Then the corresponding Coxeter graph is just $3$ isolated vertices. Moreover $m$ determines a group with a presentation: \begin{equation} \langle s_1,s_2,s_3|s_1^2=s_2^2=s_3^2=e,s_1s_2=s_2s_1,s_1s_3=s_3s_1,s_2s_3=s_3s_2\rangle \end{equation} and that is just $\mathbb{Z}_2 \times \mathbb{Z}_2 \times \mathbb{Z}_2$. Is that what they mean?