AJA Command and Staff University Journal of Systems Oriented Operations Research 2783-1493 1 3 2025 11 22 A Nonlinear Optimization Problem to Find Graphs with Maximum Energy 721345 10.22034/jmor.2024.721345 EN Sara Ahmadi Department of Mathematics, Faculty of Sciences, University of Qom, Qom, Iran 0000-0003-0127-409x Gholam Hassan Shirdel Department of Mathematics, Faculty of Sciences, University of Qom, Qom, Iran 0000-0003-2759-4606 Journal Article 2025 02 09 The energy of a graph G, denoted by E(G), is defined as the sum of the absolute values of the eigenvalues of the adjacency matrix of graph G. Until now, researchers have not been able to provide a formula for the energy of a graph in terms of graph parameters such as the number of vertices or the number of edges. Therefore, they focused on upper and lower bounds. Over the last 30 years, there has been much interest in this area, leading to the development of many new bounds. The optimal lower bound was introduced, and it was shown that star graphs have the lowest energy of all graphs without an isolated vertex. However, attempts to determine the smallest upper bound have been unsuccessful. In this paper, we take an alternative perspective, focusing on optimization, and present a nonlinear optimization problem to determine the smallest upper bound for the graph energy.