The maximum degree of Nosal graphs

A graph $G$ with $m$ edges is a Nosal graph if its spectral radius $λ(G)$ exceeds $\sqrt m$. At the threshold $λ(G)=\sqrt m$, the star $K_{1,m}$ shows that one vertex can be incident with all $m$ edges. In contrast, Li, Liu and Zhang [J. Combin. Theory Ser. B 179 (2026)] proved that every Nosal graph $G$ with sufficiently many edges satisfies $Δ(G)\le\frac12m+m^{0.99}$, where the constant $\frac12$ is best possible. In this paper, we prove that every Nosal graph $G$ with spectral radius $λ$ satisfies $Δ(G)