I am seeking proof of the following point. Any reference or direct proof would be appreciated.
Let $H$ be a directed graph, and denote by $\mathcal{H}_i$ the set of all subgraphs of $H$ with exactly $i$ vertices, consisting of a disjoint union of simple directed cycles. If $p(x)= x^n +\sum_0^{n-1}a_{k}x^k$ is the characteristic polynomial of adjagancy matrix of $H$, then for $a_i$, $$a_i=\sum_{\tilde{H}\in \mathcal{H}_{n-i}}(-1)^{c({\tilde{H}})}$$ where $c({\tilde{H}})$ is the number of cycles $\tilde{H}$ consists of.