Perron-Frobenius theorems for the numerical range of semi-monic matrix polynomials

We present an extension of the Perron-Frobenius theory to the numerical ranges of semi-monic Perron-Frobenius polynomials, namely matrix polynomials of the formQ(λ)=λmI−(λlAl+⋯+A0)=λmI−A(λ), where the coefficients are entrywise nonnegative matrices. Our approach relies on the function β↦numerical radius A(β) and the infinite graph Gm(A0,…,Al). Our main result describes the cyclic distribution of the elements of the numerical range of Q(⋅) on the circles with radius β satisfying βm=numerical radius A(β).