Extremal polyphenyl chains concerning k-matchings and k-independent sets

Denote by A(n) the set of the polyphenyl chains with n hexagons. For any A(n) is an element of A(n), let m(k)(A(n)) and m(k)(A(n)) and i(k)(A(n)) be the numbers of k-matchings and k-independent sets of A(n), respectively. In the paper, we show that for any A(n) is an element of A(n) and for any k >= 0, m(k)(M-n) <= m(k)(A(n)) <= m(k)(O-n) and i(k)(M-n) >= i(k)(A(n)) >= i(k)(O-n), with the equalities hold if A(n) = M-n or A(n) = O-n, where M-n and O-n are the meta-chain and the ortho-chain, respectively. These generalize some related results in [1].