A sharp upper bound of the nullity of a connected graph in terms of order and maximum degree

The nullity η(G) of G is the multiplicity of 0 as an eigenvalue of A(G). In this paper, we completely solve the following conjecture proposed by Zhou, Wong and Sun in [Linear Algebra and its Applications, 555 (2018) 314-320]: Let G be a connected graph of order n with nullity η(G) and the maximum degree Δ≥2. Thenη(G)≤(Δ−2)n+2Δ−1, the equality holds if and only if G≅Cn (n≡0(mod4)) or G≅KΔ,Δ.