Let $ G=(V,E) $ be a graph. A set $ F \subset E $ is called a matching if no two edges in $ F $ share a vertex. Define the matching polynomial of $ G $, $ M_{G} $ as: $$ M_{G}(x)=\sum_{F \text{ matching }} x^{|F|} $$ Show that $ M_{G} $ is real rooted.
I would appreciate any help with this. Thank you!
Hint. Start by showing that for a tree $T$ then $x^{|T|}M_T(-1/x^2)=\det(Ix-A_T)$ where $A_T$ is the adjacency matrix of $T$. Since $A_T$ is symmetric it follows that the polynomial $\det(Ix-A_T)$ is real rooted.
A useful reference can be found HERE (although the definition of matching polynomial is a bit different).