Abstract

We consider the graph $E_{3,1}$ with three generators $\sigma_1, \sigma_2, \delta$, where $\sigma_1$ has an edge with each of $\;\sigma_2$ and  $\;\delta$. We then define the Artin group of the graph $E_{3,1}$ and consider its  reduced  Perron representation of degree three. After we specialize the indeterminates used in defining the representation to  non-zero complex numbers, we obtain a necessary and sufficient condition that guarantees the irreducibility of the representation.

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