  In this paper, we introduce the notion of the quadratic graph, that is a
graph whose eigenvalues are integral or quadratic algebraic integral, and
determine nine infinite families of quadratic starlike trees, which are just
all the quadratic starlike trees including integral starlike trees. Thus the
quadratic starlike trees are completely characterized, and moreover, the
display expressions for the characteristic polynomials of the quadratic
starlike trees are also given.
