  Let $X$ be an infinite graph of bounded degree; e.g., the Cayley graph of a
free product of finite groups. If $G$ is a finite graph covered by $X$, it is
said to be $X$-Ramanujan if its second-largest eigenvalue $\lambda_2(G)$ is at
most the spectral radius $\rho(X)$ of $X$, and more generally
$k$-quasi-$X$-Ramanujan if $\lambda_k(G)$ is at most $\rho(X)$. In case $X$ is
the infinite $\Delta$-regular tree, this reduces to the well known notion of a
finite $\Delta$-regular graph being Ramanujan. Inspired by the Interlacing
Polynomials method of Marcus, Spielman, and Srivastava, we show the existence
of infinitely many $k$-quasi-$X$-Ramanujan graphs for a variety of infinite
$X$. In particular, $X$ need not be a tree; our analysis is applicable whenever
$X$ is what we call an additive product graph. This additive product is a new
construction of an infinite graph $\mathsf{AddProd}(A_1, \dots, A_c)$ from
finite 'atom' graphs $A_1, \dots, A_c$ over a common vertex set. It generalizes
the notion of the free product graph $A_1 * \cdots * A_c$ when the atoms $A_j$
are vertex-transitive, and it generalizes the notion of the universal covering
tree when the atoms $A_j$ are single-edge graphs. Key to our analysis is a new
graph polynomial $\alpha(A_1, \dots, A_c;x)$ that we call the additive
characteristic polynomial. It generalizes the well known matching polynomial
$\mu(G;x)$ in case the atoms $A_j$ are the single edges of $G$, and it
generalizes the $r$-characteristic polynomial introduced in [Ravichandran'16,
Leake-Ravichandran'18]. We show that $\alpha(A_1, \dots, A_c;x)$ is
real-rooted, and all of its roots have magnitude at most
$\rho(\mathsf{AddProd}(A_1, \dots, A_c))$. This last fact is proven by
generalizing Godsil's notion of treelike walks on a graph $G$ to a notion of
freelike walks on a collection of atoms $A_1, \dots, A_c$.
