  Recently, Gross, Mansour and Tucker introduced the partial-dual genus
polynomial of a ribbon graph as a generating function that enumerates the
partial duals of the ribbon graph by genus. It is analogous to the
extensively-studied polynomial in topological graph theory that enumerates by
genus all embeddings of a given graph. To investigate the partial-dual genus
polynomial one only needs to focus on bouquets, i.e. ribbon graphs with only
one vertex. In this paper, we shall further show that the partial-dual genus
polynomial of a bouquet essentially depends on the signed intersection graph of
the bouquet rather than on the bouquet itself. That is to say the bouquets with
the same signed intersection graph will have the same partial-dual genus
polynomial. We then prove that the partial-dual genus polynomial of a bouquet
contains non-zero constant term if and only if its signed intersection graph is
positive and bipartite. Finally we consider a conjecture posed by Gross,
Mansour and Tucker. that there is no orientable ribbon graph whose partial-dual
genus polynomial has only one non-constant term, we give a characterization of
non-empty bouquets whose partial-dual genus polynomials have only one term by
consider non-orientable case and orientable case separately.
