  We provide a sufficient condition for the numerical range of a nilpotent
matrix N to be circular in terms of the existence of cycles in an undirected
graph associated with N. We prove that if we add to this matrix N a diagonal
real matrix D, the matrix D + N has convex numerical range. For 3 x 3 nilpotent
matrices, we strength further our results and obtain necessary and sufficient
conditions for circularity and convexity of the numerical range.
