TITLE: Intuition of the energy of a graph
QUESTION [7 upvotes]: The energy of a a simple graph $G$ is defined to be the sum of the absolute values of the eigenvalues of $G$. What is a good intuition of it?

REPLY [2 votes]: The energy of a graph equals to the nuclear norm $\|A\|_*$ of the adjacency matrix $A$ of the graph.

So, an intuition of the nuclear norm can give an intuition of the energy of a graph.
Therefore answers to the following question may be useful:

What is the intuition for the trace norm (nuclear norm)?

The following intuition may be more interesting which has some "energy" sense
$$
\|A\|_{\mathrm{*}} = \inf\big\{ \sum_j \|X_j\| : A = \sum_j X_j, \ \ \mathrm{rank}(X_j) = 1 \ \ \forall j \big\}.
$$