# Mathematical Energy: Etymology

This might be the silliest post I’ve written so far (yes – worse than “Was Disney trying to kill mathematicians?”). I urge you to stop reading now unless (i) you are quite familiar with the mathematical notion of energy (e.g., additive energy), and (ii) you have a horrible sense of humor.

The term energy was coined by Tao and Vu. I like “energy” as a name for this object, but I never had a good answer when asked why this is how it’s called. That is, until a referee report provided me with an answer. And this wonderful referee probably didn’t even know it.

In a recent paper, Cosmin Pohoata and I used “color energy”. We have a graph $G=(V,E)$ with colored edges. Denote the color of an edge $(v_1,v_2)$ as $\chi(v_1,v_2)$. The color energy of $G$ is

$E(G) = \left|\left\{(v_1,v_2,v_3,v_4)\in V^4 :\ \chi(v_1,v_2) = \chi(v_3,v_4) \right\}\right|.$

The referee complained about our notation for the multiplicity of a color $c$ (the number of edge of color $c$), and asked to change it to $m_c$. After this change of notation, the energy is defined by the standard formula

$E = \sum m_c^2.$