Exact variance-energy relations and optimal spectral bounds for graphs

Let \(G\) be a graph of order \(n\) and size \(m\), with adjacency matrix eigenvalues \(λ_1 ≥ λ_2 ≥ ≥ λ_n\). 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 \(x = (|λ_1|, |λ_2|, , |λ_n|)\,.\) Specifically, we prove that \(E(G) = √2mn - n^2 Var(x),\) 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 \(G K_n\).