Bounds for the Energy of Graphs.

Let G be a graph on n vertices and m edges, with maximum degree Δ (G) and minimum degree δ (G). Let A be the adjacency matrix of G, and let λ 1 ≥ λ 2 ≥ ... ≥ λ n be the eigenvalues of G. The energy of G, denoted by E (G) , is defined as the sum of the absolute values of the eigenvalues of G, that is E (G) = | λ 1 | + ... + | λ n | . The energy of G is known to be at least twice the minimum degree of G, E (G) ≥ 2 δ (G). Akbari and Hosseinzadeh conjectured that the energy of a graph G whose adjacency matrix is nonsingular is in fact greater than or equal to the sum of the maximum and the minimum degrees of G, i.e., E (G) ≥ Δ (G) + δ (G). In this paper, we present a proof of this conjecture for hyperenergetic graphs, and we prove an inequality that appears to support the conjectured inequality. Additionally, we derive various lower and upper bounds for E (G) . The results rely on elementary inequalities and their application. [ABSTRACT FROM AUTHOR]