Abstract
The energy of a graph \(G\), denoted by \(\mathcal{E}(G)\), is defined as the sum of the absolute values of all eigenvalues of \(G\). It is proved in [MATCH Commun. Math. Comput. Chem. 79 (2018) 287--301] by Alawiah et al. that \(\mathcal{E}(G)\leq 2\sqrt{\Delta}+\sqrt{(n - 2)( 2m - 2\Delta)}\) for every bipartite graph \(G\) of order \(n\), size \(m\) and maximum degree \(\Delta\). We prove the above bound for all graphs \(G\). We also prove new types of two bounds of Koolen and Moulton given in [Adv. Appl. Math. 26 (2001) 47-52] and [Graphs Comb. 19 (2003) 131-135].
