Assume there are N countries. The cost of making a phone call from country $i$ to country $j$ is $C_{ij}$. We know that all costs are non-negative.
(Q1) Can you think of a verbal interpretation of eigenvalues of the matrix $C_{ij}$?
(Q2) Does anything change, if we allow weights to be negative?
I am aware that an eigendecomposition of a transformation $T$ is given by $T = R^{-1}DR$, which means that, if a matrix were to be used as a transformation, it could be interpreted as rotation, scaling, and rotation back to the original basis. However, I'm not necessarily using my matrix to transform anything, so my intuition does not quite help
You may want to take a look at eigenvector centrality. This states that the centrality of node $i$ is the weighted average of the centrality of nodes it is connected to: \begin{align} v_i &= \frac{1}{\beta}\sum_{j\in N}C_{ij}v_j\\ \implies \beta\vec{v} &=C\vec{v} \end{align}
Thus the eigenvalue can be interpreted as the *inverse of the weights/frequencies that each node attaches to its neighbor's centralities. Mathematically, though, the eigenvalues determines the spectrum of the graph. Naively speaking, larger eigenvalues should correspond to more compact graphs (e.g. stars).