Extremal Polygonal Chains Concerning Merrified-Simmons Index

Denote by An the set of h-polygonal chains with n congruent regular h-polygons (h >= 6). For any A(n) is an element of A(n), let i(k) (A(n)) be the numbers of k-independent sets of A(n). In this article, we show that i(k) (Z(n)(2)) >= i(k) (A(n)) >= i(k) (Z(n)(1)) for any k >= 0, with the equalities on the left holding for all k only if A(n) = Z(n)(2), and the equalities on the right holding for all k only if A(n) = Z(n)(1), where Z(n)(1) and Z(n)(2) are extremal chains of type one and type two (their definitions are given in the main text), respectively. Thus, we extend the main results of (7) to a more general case.