Title: Graphical virtual links and a polynomial of signed cyclic graphs,
Abstract: For a signed cyclic graph G, we can construct a unique virtual link L by
taking the medial construction and convert 4-valent vertices of the medial
graph to crossings according to the signs. If a virtual link can occur in this
way then we say that the virtual link is graphical. In the article we shall
prove that a virtual link L is graphical if and only if it is checkerboard
colorable. On the other hand, we introduce a polynomial F[G] for signed cyclic
graphs, which is defined via a deletion-marking recursion. We shall establish
the relationship between F[G] of a signed cyclic graph G and the bracket
polynomial of one of the virtual link diagrams associated with G. Finally we
give a spanning subgraph expansion for F[G].