  Recently, Choi and Park introduced an invariant of a finite simple graph,
called signed a-number, arising from computing certain topological invariants
of some specific kinds of real toric manifolds. They also found the signed
a-numbers of path graphs, cycle graphs, complete graphs, and star graphs.
  We introduce a signed a-polynomial which is a generalization of the signed
a-number and gives a-, b-, and c-numbers. The signed a-polynomial of a graph
$G$ is related to the Poincar\'e polynomial $P_{M(G)}(z)$, which is the
generating function for the Betti numbers of the real toric manifold $M(G)$. We
give the generating functions for the signed a-polynomials of not only path
graphs, cycle graphs, complete graphs, and star graphs, but also complete
bipartite graphs and complete multipartite graphs. As a consequence, we find
the Euler characteristic number and the Betti numbers of the real toric
manifold $M(G)$ for complete multipartite graphs $G$.
