Let me start with some definitions and notations.
Definition: The graph of a symmetric matrix $A=[a_{ij}]_{i,j=1}^{n}$ is defined to be a simple graph $G$ where, $i\sim j$ in $G$ if $a_{ij}\neq 0$, for $i\neq j$.
It is clear that the graph of $A$ does not depend on the diagonal entries of $G$.
Notation: Assume that graph of $A$ is a tree $T$ and let $T(i)$ denote the graph obtained from $T$ by deleting vertex $i$, and for $j\sim i$ let $T_j(i)$ denote the subgraph of $T(i)$ which contains the vertex $j$. Here is an example:
Similarly, let $A(i)$ denote the matrix obtained from $A$ by deleting the $i$-th row and column, and let $A_j(i)$ denote the submatrix of $A$ corresponding to the vertices in $T_j(i)$. Also, if $B$ is a collection of vertices, then $A[B]$ denotes the submatrix of $A$ corresponding to those vertices of $T$.
There are many formulas for different expansions of the characteristic polynomial of $A$, $c_A(x)$, when its graph is a tree. For example, one can easily see that $$c_A = a_{ii} c_{A(i)} - \sum_{ j\sim i} a_{ij}^2 c_{A_j(i)}$$
The easy way to see this equality is by expanding (Laplace expansion) the determinant of $xI-A$ over the $i$-th row. But in many other cases, a simple Laplace expansion is not enough. For example, if graph of $A$ is an tree $T$ (below) and $r$ and $s$ are adjacent, then we have the following:
$$c_A = -a_{rs}^2 c_{A[V_r\setminus \{r\}]} c_{A[V_s\setminus \{s\}]} + c_{A[V_r]} c_{A[V_s]}.$$
Fact: There is a very common treatment of these equations in literature by considering the correspondence between the nonzero terms of the characteristic polynomial of $A$ in the following expansion, and the cycle covers of directed graph of $A$ which is obtained from $T$ by replacing each edge of it with two directed edges in opposite directions and adding a loop to each vertex.
$$c_A(x) = \sum_{\sigma \in S_n} (-1)^{sign(\sigma)} \prod_{i=1}^{n} B_{i\sigma(i)}, $$ where $B=xI-A$.
Question: I need to use this, but I have never seen a book or article that mentions this correspondence formally. Do you know any such references? Maybe any works by Brualdi, Ryser, Godsil, Biggs, Cvetkovic etc?
First figure is extracted from: http://www.sciencedirect.com/science/article/pii/S0024379513001006?np=y