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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.

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