Theorem 4.22 (Godsil-Gutman, 80s).

$${𝔼}_{S\sim \{\pm 1\}}\det (xI-A_s)={\mu }_G(x),$$

where

$${\mu }_G(x)=\sum _{k\ge 0}{x}^{n-2k}{(-1)}^k{m}_k(G),$$

where $m_k(G)$ is the number of matchings in $G$ with $k$ edges.