  Let $G=(V,E)$ be a simple connected graph. A matching of $G$ is a set of
disjoint edges of $G$. For every $n, m\in\mathbb{N}$, the $n$-subdivision of
$G$ is a simple graph $G^{\frac{1}{n}}$ which is constructed by replacing each
edge of $G$ with a path of length $n$ and the $m$th power of $G$, denoted by
$G^m$, is a graph with the same vertex set as $G$ such that two vertices are
adjacent in $G^m$ if and only if their distance is at most $m$ in $G$. The
$m^{th}$ power of the $n$-subdivision of $G$ has been introduced as a
fractional power of $G$ and is denoted by $G^{\frac{m}{n}}$. In this paper, we
study some parameters related to matching of the natural and the fractional
powers of some specific graphs. Also we study these parameters for power of
graphs that are importance of in Chemistry.
