Given a graph $G$ with $n$ vertices, if its adjacency matrix $A$ has eigenvalues $\lambda_1 \geq \lambda_2 \geq . . . \geq \lambda_n$ then the energy is defined as: $$\mathcal{E}(G) =\sum_{i=1}^{n} |\lambda_i|.$$
By trials and errors experimentation, with a simple undirected edge weighted path T, it seems that the energy of $T$., or $\mathcal{E}(T)$ proportionally increases with increasing weight of $T$ (but doesn't seem to hold in general graph).
How to prove that $\mathcal{E}(T)$ increases with increasing weight of $T$?