On the number of perfect matchings of middle graphs

Let $G$ be a connected graph with $n$ vertices and $m$ edges, and let $L\left(G\right)$ and $M\left(G\right)$ be the line graph and middle graph of $G$, respectively. Let $p\left(G\right)$ be the number of perfect matchings of $G$. Dong, Yan and Zhang (Discrete Applied Mathematics, 161 (2013) 794-801.) proved that if $m$ is even, then $p\left(L\left(G\right)\right)\ge 2^{m-n+1}$, and characterized the extremal graphs $G$ with equality. In this paper, we obtain a similar result for the middle graph of $G$, that is, if $m+n$ is even, then $p\left(M\left(G\right)\right)\ge 2^{m-n+1}{3}^{\frac{m-n}{2}}$, and characterize the extremal graphs $G$ such that $p\left(M\left(G\right)\right)=2^{m-n+1}{3}^{\frac{m-n}{2}}$. As applications, we enumerate perfect matchings of the middle graphs of the Cartesian products $P_n\times P_m,C_n\times P_m$, and $C_n\times C_m$, where $P_n$ and $C_n$ are the path and cycle with $n$ vertices.