Let \ (G\) be a graph of order \ (n\) and size \ (m\), with adjacency matrix eigenvalues \ (₁ ₂ ₙ\). The energy of \ (G\), denoted by \ (E (G) \), is defined as the sum of the absolute values of its eigenvalues. A classical upper bound on the energy, originally established by McClelland 1, states that \ (E (G) 2mn\,. \) In this paper, we refine the spectral analysis of graph energy by deriving an exact analytical expression relating \ (E (G) \) to the variance of the vector of absolute eigenvalues \ (x = (|₁|, |₂|, , |ₙ|) \,. \) Specifically, we prove that \ (E (G) = 2mn - n² Var (x), \) providing a more precise and quantitative spectral characterization of graph energy. As an application, this identity allows us to derive improved lower bounds for \ (E (G) \), thereby strengthening and generalizing previously known inequalities. Furthermore we conjecture that for any non-singular graph \ (G\) of order \ (n\), \ (E (G) 2 d (n-1), \) where \ (d = 2m/n\) is the average vertex degree of \ (G\). Equality holds if and only if \ (G Kₙ\).
Jahanbani et al. (Thu,) studied this question.