On Topological Indices and Arithmetic–Geometric Energy of Graphs

Let Γ be a graph of order n with m edges, and let Aag(Γ) denote its arithmetic–geometric matrix. The eigenvalues of Aag(Γ) are referred to as the arithmetic–geometric eigenvalues of Γ. The sum of the absolute values of these eigenvalues is called the arithmetic–geometric energy of Γ. In this paper, we establish some new upper and lower bounds for the arithmetic–geometric energy in terms of various topological indices, and we completely characterize the extremal graphs attaining these bounds. We show that our results significantly improve upon several existing bounds reported in the literature. We also provide several ways to construct graphs with the same AM–GM energy but different AM–GM spectra.