Extremal Polyphenyl Chains Concerning $k$-Matchings and $k$-Independent Sets

Abstract

Denote by ${\mathcal{A}}_{\mathcal{n}}$, the set of the polyphenyl chains with $n$ hexagons. For any $A_n\in {\mathcal{A}}_{\mathcal{n}}$, let $m_k(A_n)$ 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 $A_n\in {\mathcal{A}}_{\mathcal{n}}$ and for any $k\ge 0$,$m_k(M_n)\le m_k(A_n)\le m_k(O_n)\text{and}{i}_k(M_n)\ge i_k(A_n)\ge i_k(O_n),$ with the equalities holding 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 $[1]$.