Permanents of Hexagonal and Armchair Chains

The permanent is important invariants of a graph with some applications in physics. If $G$ is a graph with adjacency matrix $A=\left[a_{ij}\right]$ , then the permanent of $A$ is defined as $\text{perm}\left(A\right)={\displaystyle {\sum }_{\sigma \in S_n}{\displaystyle {\prod }_{i=1}^n{a}_{i\sigma \left(i\right)}}}$ , where $S_n$ denotes the symmetric group on $n$ symbols. In this paper, the general form of the adjacency matrices of hexagonal and armchair chains will be computed. As a consequence of our work, it is proved that if $G\left[k\right]$ and $H\left[k\right]$ denote the hexagonal and armchair chains, respectively, then $\text{perm}\left(A\left(G\left[1\right]\right)\right)=4$ , $\text{perm}\left(A\left(G\left[k\right]\right)\right)={\left(k+1\right)}^2$ , $k\ge 2$ , and $\text{perm}\left(A\left(H\left[k\right]\right)\right)=4^k$ with