Let G be a graph with n vertices and m edges. The spectral radius \(\rho (G)\) of G is the largest eigenvalue of the adjacency matrix of G. It is well-known that \(\rho (G)\ge \frac{2m}{n}\) with equality if and only if G is regular. To bound \(\rho (G)-\frac{2m}{n}\) , Nikiforov (2006) introduced the degree deviation of G as \(\begin{aligned} s(G)=\sum _{1\le i\le n}\left| d_{i}-\frac{2\,m}{n}\right| , \end{aligned}\) where \(d_{1},d_{2},\ldots ,d_{n}\) are the degrees of the vertices of G. Nikiforov conjectured that \(\rho (G)-\frac{2m}{n}\le \sqrt{\frac{1}{2}s(G)}\) for sufficiently large m and n. In this paper, we settle this conjecture without the assumption that m and n are large.
