The energy of a graph is defined as the sum of the absolute values of its graph eigenvalues (i.e., the sum of its graph spectrum
terms).
Other varieties of graph energy are defined analogously using different matrices associated with a graph (and in particular, a weighted
adjacency matrix).
As summarized by Alikhani and Ghanbari (2024), the energy of a graph cannot be an odd integer (Bapat and Pati 2004), the square root of an odd integer (Pirzada and
Gutman 2008), or the golden ratio (Alikhani and Iranmanesh
2010).
Alikhani, S. and Ghanbari, N. "Golden Ratio in Graph Theory: A Survey." 9 Jul 2024. https://arxiv.org/abs/2407.15860.Alikhani,
S. and Iranmanesh, M. A. "Energy of Graphs, Matroids and Fibonacci Numbers."
Iranian J. Math. Sci. Inform. 5, 55-60, 2010.Bapat, R. B.
and Pati, S. "Energy of a Graph Is Never an Odd Integer." Bull. Kerala
Math. Assoc.1, 129-132, 2004.Cvetković, D. M.;
Doob, M.; Sachs, H. Spectra
of Graphs. New York: Academic Press, 1980.Gutman, I. "The
Energy of a Graph." In 10. Steiermärkisches Mathematisches Symposium
(Stift Rein, Graz, 1978).Ber. Math.-Statist. Sekt. Forsch. Graz103,
1-22, 1978.Gutman, I. "The Energy of a Graph: Old and New Results."
In Algebraic Combinatorics and Applications (Gößweinstein, 1999).
Berlin: Springer, pp. 196-211, 2001.Li, X.; Shi, Y.; and Gutman, I.
Graph
Energy. New York: Springer, 2012.Pirzada, S. and Gutman, I.
"Energy of Graph Is Never the Square Root of an Odd Integer." Appl.
Anal. Disc. Math.2, 118-121, 2008.
