From the abstract: It is proven in [\textit{M. de Bondt, A. van den Essen}, Proc. Am. Math. Soc. 133, No.~8, 2201--2205 (2005; Zbl 1073.14077)] that it suffices to study the Jacobian conjecture for maps of the form \(x+\nabla f\), where \(f\) is a homogeneous polynomial of degree \(d=4\). The Jacobian conjecture implies that \(f\) is a finite sum of \(d\)-th powers of linear forms \(^d\) where \(=x^ty\) and each \(\alpha\) is an isotropic vector i.e. \(=0\). To a set \(\{\alpha_1,\dots, \alpha_s\}\) of isotropic vectors, we assing a graph and study its structure in case the corresponding polynomial \(f=\sum ^d\) has a nilpotent Hessian. The main result of this article asserts that in the case \(\dim([\alpha_1,\dots, \alpha_s])\leq 2\) or \(\geq s-2\), the Jacobian conjecture holds for maps \(x+\nabla f\). In fact, we give a complete description of the graphs of such \(f\)'s, whose Hessian is nilpotent. As an application of the result, we show that lines and cycles cannot appear as graphs of HN polynomials.