A Perron number is an algebraic number which is greater than one in absolute value and is greater than all of its Galois conjugates in absolute value as well. Lind's theorem states that any Perron number is spectral radius of some Perron-frobenius matrix (a matrix $A$ is Perron-Frobenius if all of its entries are non-negative integers and there is some $n>0$ such that all entries of $A^n$ are positive).
Question: Given a Perron number $\lambda$, is there any lower bound for the size of smallest Perron-Frobenius matrix $A$ with spectral radius equal to $\lambda$ (called Perron-frobenius degree)? Besides algebraic degree off course
PS: In particular I'm interested in Perron numbers which come from a topological construction, namely as stretch factors of pseudo-Anosov homeomorphisms of hyperbolic surfaces. There is an upper bound for size of such matrix , 6g-6 where g is the genus of closed surface, however I'm not aware of lower bounds.