  An edge subset $S$ of a connected graph $G$ is called an anti-Kekul\'{e} set
if $G-S$ is connected and has no perfect matching. We can see that a connected
graph $G$ has no anti-Kekul\'{e} set if and only if each spanning tree of $G$
has a perfect matching. In this paper, by applying Tutte's 1-factor theorem and
structure of minimally 2-connected graphs, we characterize all graphs whose
each spanning tree has a perfect matching In addition, we show that if $G$ is a
connected graph of order $2n$ for a positive integer $n\geq 4$ and size $m$
whose each spanning tree has a perfect matching, then $m\leq \frac{(n+1)n} 2$,
with equality if and only if $G\cong K_n\circ K_1$.
