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).