Extremal Polygonal Chains Concerning Merrified-Simmons Index

ABSTRACTDenote by the set of h-polygonal chains with n congruent regular h-polygons (h ⩾ 6). For any , let ik(An) be the numbers of k-independent sets of An. In this article, we show that ik(Z2n) ⩾ ik(An) ⩾ ik(Z1n) for any k ⩾ 0, with the equalities on the left holding for all k only if An = Z2n, and the equalities on the right holding for all k only if An = Z1n, where Z1n and Z2n 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.