Bounds for the Energy of Graphs

Let G be a graph on n vertices and m edges, with maximum degree Delta(G) and minimum degree delta(G). Let A be the adjacency matrix of G, and let lambda(1)>=lambda(2)>= ,,, >=lambda(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)=|lambda(1)|+ ... +|lambda(n)|. The energy of G is known to be at least twice the minimum degree of G, E(G)>= 2 delta(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)>=Delta(G)+delta(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.