The term energy has an important physical meaning in physics and is an extremely useful concept. There are several forms energy defined in mathematics.
In measure theory, let 
be a space with measure 
and let 
be a real function on the product
space 
. When
 |  |  |
(1)
|
 |  |  |
(2)
|
exists for measures 
,
is called the mutual
energy and 
is called the energy (Iyanaga and Kawada 1980, p. 1038)
In harmonic function theory, let 
be a real-valued harmonic
function on a bounded domain Omega, then the Dirichlet
energy is defined as 
,
where 
is the gradient.
In graph theory, graph energy is defined as the sum of absolute values of the graph eigenvalues (i.e., eignvalues of a graph's adjacency matrix). Other varieties of graph energy are defined analogously using different matrices associated with a graph (and in particular, a weighted adjacency matrix).
