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] that \( \mathcal{E} (G)\leq 2+\sqrt{(n-1)(2m-4)} \) if \( G \) is a connected unicyclic graph. We prove a generalization of the above bound for all graphs \( G \). We then prove a new sharp upper bound for the energy of bipartite graphs, and in particular we improve the famous bound \( \mathcal{E} (G)\leq \frac{n}{\sqrt{8}}(\sqrt{n}+\sqrt{2}) \) of Koolen and Moulton on bipartite graphs given in [Graphs Combin. 19 (2003), 131--135] under certain conditions. We also prove upper and lower bounds for the energy of graphs arisen by the Mycielski construction.
