فهرست نشریات علمی دسترسی آزاد ایران

‎This paper proposes several new statistical bounds for graph energy derived from the eigenvalues of the adjacency matrix‎. ‎Using inequalities involving the arithmetic‎, ‎geometric‎, ‎and generalized means‎, ‎along with variance and standard deviation‎, ‎we establish both upper and lower bounds for $E(G)$‎. ‎These statistical bounds capture not only mean relationships but also eigenvalue variability‎, ‎offering more flexible and accurate estimates than conventional deterministic inequalities‎. ‎The approach integrates tools from inequality theory and spectral graph theory‎, ‎with applying weighted means and Jensen-type inequalities‎. ‎We also conjecture based on numerical evidence that the energy-to-geometric mean ratio converges to a constant value for large Erd\"{o}s-R\'{e}nyi random graphs‎. ‎A detailed analysis of path graphs demonstrates the effectiveness of the proposed bounds‎, ‎offering improved estimates‎.