On the rank (nullity) of a connected graph

The rank $r(G)$ of a graph $G$ is the rank of its adjacency matrix $A(G)$ and the nullity $\eta(G)$ of $G$ is the multiplicity of $0$ as an eigenvalue of $A(G)$. In this paper, we prove that if $G$ is a connected graph of order $n$ with rank $r$, then $G$ contains a nonsingular connected induced subgraph of order $r$. As an application of the result, we completely solve the following problem posed 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 $\eta(G)$ and the maximum degree $\Delta$. Then $$\eta(G)\le\frac{(\Delta-2)n+2}{\Delta-1},$$ the equality holds if and only if $G\cong C_n$ ($n\equiv 0$ $(mod\ 4)$) or $G\cong K_{\Delta, \Delta}$.