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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?

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The name "energy" only makes physical sense for a bipartite graph, where the eigenvalues of the adjacency matrix come in pairs $\pm\lambda$. If graph represents a molecule, the adjacency matrix is the Hamiltonian in the socalled tight-binding approximation. (The vertices are atomic orbitals and the edges are nearest-neigbor bonds.) The binding energy of the molecule is the sum of all the positive eigenvalues, which equals (up to a factor of two) the sum of the absolute values. If you have intuition for which chemical bonds would destabilize a molecule, by lowering its binding energy, then the energy interpretation of $\sum|\lambda|$ can be helpful, but if you lack chemical intuition then I'm not sure that this interpretation is helpful.

When the graph is not bipartite the graph energy has no physical or chemical meaning. The sum $\sum|\lambda|$ remains a useful way to characterize a graph, see for example What is the meaning of the graph energy after all? (2017).

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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 $$ \|A\|_{\mathrm{*}} = \inf\big\{ \sum_j \|X_j\| : A = \sum_j X_j, \ \ \mathrm{rank}(X_j) = 1 \ \ \forall j \big\}. $$

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