$G$ may have loops and multiple, directed edges, and we take $H\subset G$ to mean that the edges and vertices of $H$ are subsets of those in $G$ (not necessarily a vertex-induced subgraph).
We take $\rho(G)$ to be the spectral radius of $G$; that is, the largest absolute value of the eigenvalues of its adjacency matrix. From the Perron-Froebenius theorem, $\rho(G)$ is itself an eigenvalue with an eigenvector that has all positive components.
I was trying to prove the above for $n=2$ (and on graphs satisfying an additional condition) to prove a relationship between indefinite generalized Cartan matrices and affine Cartan matrices, but then began to believe it might be true more generally.
Edit for clarification: $\rho(G)$ needn't be an integer, but $n$ is