Graph energy

In mathematics, the energy of a graph is the sum of the absolute values of the eigenvalues of the adjacency matrix of the graph. This quantity is studied in the context of spectral graph theory.

More precisely, let G be a graph with n vertices. It is assumed that G is simple, that is, it does not contain loops or parallel edges. Let A be the adjacency matrix of G and let $\lambda_i$, $i = 1 , \ldots , n$, be the eigenvalues of A. Then the energy of the graph is defined as:

$E(G) = \sum_{i=1}^n|\lambda_i|.$

References

• Cvetković, Dragoš M.; Doob, Michael; Sachs, Horst (1980), Spectra of graphs, Pure and Applied Mathematics 87, New York: Academic Press Inc. [Harcourt Brace Jovanovich Publishers], ISBN 0-12-195150-2, MR 572262.
• Gutman, Ivan (1978), "The energy of a graph", 10. Steiermärkisches Mathematisches Symposium (Stift Rein, Graz, 1978), Ber. Math.-Statist. Sekt. Forsch. Graz 103, pp. 1–22, MR 525890.
• Gutman, Ivan (2001), "The energy of a graph: old and new results", Algebraic combinatorics and applications (Gößweinstein, 1999), Berlin: Springer, pp. 196–211, MR 1851951.

• Li, Xueliang; Shi, Yongtang; Gutman, Ivan (2012), Graph Energy, New York: Springer, ISBN 978-1-4614-4219-6.