Extremal hexagonal chains with respect to the coefficients sum of the permanental polynomial

A hexagonal system is a graphical representation of a benzenoid hydrocarbon in theoretical chemistry. A hexagonal chain is a cata-condensed hexagonal system with no branchings. In this paper we consider extremal hexagonal chains with respect to the coefficients sum of the permanental polynomial. We prove that the linear chain attains the minimum value of this sum and the zigzag chain attains the maximum value of this sum.