mathematics Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...

, the energy of a

graph Graph may refer to: Mathematics *Graph (discrete mathematics), a structure made of vertices and edges **Graph theory, the study of such graphs and their properties *Graph (topology), a topological space resembling a graph in the sense of discret ...

is the sum of the

absolute value In mathematics, the absolute value or modulus of a real number x, is the non-negative value without regard to its sign. Namely, , x, =x if x is a positive number, and , x, =-x if x is negative (in which case negating x makes -x positive), ...

s of the

eigenvalue In linear algebra, an eigenvector ( ) or characteristic vector is a vector that has its direction unchanged (or reversed) by a given linear transformation. More precisely, an eigenvector \mathbf v of a linear transformation T is scaled by a ...

s of the

adjacency matrix In graph theory and computer science, an adjacency matrix is a square matrix used to represent a finite graph (discrete mathematics), graph. The elements of the matrix (mathematics), matrix indicate whether pairs of Vertex (graph theory), vertices ...

of the graph. This quantity is studied in the context of

spectral graph theory In mathematics, spectral graph theory is the study of the properties of a Graph (discrete mathematics), graph in relationship to the characteristic polynomial, eigenvalues, and eigenvectors of matrices associated with the graph, such as its adjacen ...

. More precisely, let ''G'' be a graph with ''n'' vertices. It is assumed that ''G'' is a

simple graph In discrete mathematics, particularly in graph theory, a graph is a structure consisting of a Set (mathematics), set of objects where some pairs of the objects are in some sense "related". The objects are represented by abstractions called ''Ver ...

, that is, it does not contain loops or parallel edges. Let ''A'' be the

of ''G'' and let

\lambda_i

i = 1 , \ldots , n

, be the eigenvalues of ''A''. Then the energy of the graph is defined as: :

E(G) = \sum_^n, \lambda_i, .

References

*. *. *. *. Algebraic graph theory {{graph-stub